TPTP Problem File: SLH0972^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0024_Restrict_Frees/prob_00098_003850__17780014_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1752 ( 590 unt; 475 typ; 0 def)
% Number of atoms : 4128 (1427 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 13970 ( 662 ~; 134 |; 412 &;10992 @)
% ( 0 <=>;1770 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 71 ( 70 usr)
% Number of type conns : 1325 (1325 >; 0 *; 0 +; 0 <<)
% Number of symbols : 408 ( 405 usr; 48 con; 0-4 aty)
% Number of variables : 4125 ( 582 ^;3425 !; 118 ?;4125 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:27:11.622
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member4819239714442960612at_nat: produc2625736947911470797at_nat > set_Pr2403587099507754371at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
member6199062629724834769at_nat: produc3676383722749475752at_nat > set_Pr8118092986756011272at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
member5724791357406019196od_o_o: produc3531288590874529381od_o_o > set_Pr4444513774987008283od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member4689715034089797240at_nat: produc364298880807886031at_nat > set_Pr3205448351804440111at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3635249065003913160la_a_b: produc8504780217128037151la_a_b > set_Pr389361089297601407la_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member6956540099943067662at_nat: produc5825016348098550007at_nat > set_Pr2645174627780777389at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
member8660133138740149732at_nat: produc3186665129434246203at_nat > set_Pr6517276030303973275at_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_v__092_060Q_062,type,
q: set_Pr2645174627780777389at_nat ).
% Relevant facts (1275)
thf(fact_0_case__prodI,axiom,
! [F: $o > $o > $o,A: $o,B: $o] :
( ( F @ A @ B )
=> ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) ) ) ).
% case_prodI
thf(fact_1_case__prodI,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( F @ A @ B )
=> ( produc5861013030283631554_nat_o @ F @ ( produc760948067230524457at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_2_case__prodI2,axiom,
! [P: product_prod_o_o,C: $o > $o > $o] :
( ! [A2: $o,B2: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( C @ A2 @ B2 ) )
=> ( produc6197397395684419436_o_o_o @ C @ P ) ) ).
% case_prodI2
thf(fact_3_case__prodI2,axiom,
! [P: produc5825016348098550007at_nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( C @ A2 @ B2 ) )
=> ( produc5861013030283631554_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_4_case__prod__conv,axiom,
! [F: $o > $o > $o,A: $o,B: $o] :
( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_5_case__prod__conv,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( produc5861013030283631554_nat_o @ F @ ( produc760948067230524457at_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6_fixfree__def,axiom,
( restrict_fixfree_a_b
= ( ^ [Q_fin: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [Qfix: relational_fmla_a_b,Qeq: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ Qfix @ Qeq ) @ Q_fin )
& ( ( relati62690040636126068ns_a_b @ Qfix )
!= bot_bot_set_nat ) ) ) ) ) ) ).
% fixfree_def
thf(fact_7_split__part,axiom,
! [P2: $o,Q: $o > $o > $o] :
( ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] :
( P2
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: product_prod_o_o] :
( P2
& ( produc6197397395684419436_o_o_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_8_split__part,axiom,
! [P2: $o,Q: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( produc5861013030283631554_nat_o
@ ^ [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat] :
( P2
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: produc5825016348098550007at_nat] :
( P2
& ( produc5861013030283631554_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_9_finite__Collect__conjI,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P2 ) )
| ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ Q ) ) )
=> ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_10_finite__Collect__disjI,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P2 ) )
& ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_11_empty__iff,axiom,
! [C: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ C @ bot_bo1973934853101755969at_nat ) ).
% empty_iff
thf(fact_12_empty__iff,axiom,
! [C: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ C @ bot_bo7073875226086086771od_o_o ) ).
% empty_iff
thf(fact_13_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_14_all__not__in__conv,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( ! [X: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ X @ A4 ) )
= ( A4 = bot_bo1973934853101755969at_nat ) ) ).
% all_not_in_conv
thf(fact_15_all__not__in__conv,axiom,
! [A4: set_Product_prod_o_o] :
( ( ! [X: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ X @ A4 ) )
= ( A4 = bot_bo7073875226086086771od_o_o ) ) ).
% all_not_in_conv
thf(fact_16_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_17_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_18_Collect__empty__eq,axiom,
! [P2: product_prod_o_o > $o] :
( ( ( collec3167064739498627218od_o_o @ P2 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X: product_prod_o_o] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_19_Collect__empty__eq,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
( ( ( collec7697611494764367948at_nat @ P2 )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P2: product_prod_o_o > $o] :
( ( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o @ P2 ) )
= ( ! [X: product_prod_o_o] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_22_empty__Collect__eq,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
( ( bot_bo1973934853101755969at_nat
= ( collec7697611494764367948at_nat @ P2 ) )
= ( ! [X: produc5825016348098550007at_nat] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A5: relational_fmla_a_b,B4: set_Pr1261947904930325089at_nat] :
( ( ( produc760948067230524457at_nat @ A @ B )
= ( produc760948067230524457at_nat @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A: $o,B: $o,A5: $o,B4: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_25_prod_Oinject,axiom,
! [X1: relational_fmla_a_b,X2: set_Pr1261947904930325089at_nat,Y1: relational_fmla_a_b,Y2: set_Pr1261947904930325089at_nat] :
( ( ( produc760948067230524457at_nat @ X1 @ X2 )
= ( produc760948067230524457at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_26_prod_Oinject,axiom,
! [X1: $o,X2: $o,Y1: $o,Y2: $o] :
( ( ( product_Pair_o_o @ X1 @ X2 )
= ( product_Pair_o_o @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_27_mem__case__prodI2,axiom,
! [P: produc5825016348098550007at_nat,Z: produc5825016348098550007at_nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Pr2645174627780777389at_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( member6956540099943067662at_nat @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member6956540099943067662at_nat @ Z @ ( produc4831187842317280515at_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_28_mem__case__prodI2,axiom,
! [P: produc5825016348098550007at_nat,Z: product_prod_o_o,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Product_prod_o_o] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( member7466972457876170832od_o_o @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member7466972457876170832od_o_o @ Z @ ( produc1858041438500342065od_o_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_29_mem__case__prodI2,axiom,
! [P: produc5825016348098550007at_nat,Z: nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( member_nat @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member_nat @ Z @ ( produc6938862770440504796et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_30_mem__case__prodI2,axiom,
! [P: product_prod_o_o,Z: produc5825016348098550007at_nat,C: $o > $o > set_Pr2645174627780777389at_nat] :
( ! [A2: $o,B2: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( member6956540099943067662at_nat @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member6956540099943067662at_nat @ Z @ ( produc3500709797993629913at_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_31_mem__case__prodI2,axiom,
! [P: product_prod_o_o,Z: product_prod_o_o,C: $o > $o > set_Product_prod_o_o] :
( ! [A2: $o,B2: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( member7466972457876170832od_o_o @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member7466972457876170832od_o_o @ Z @ ( produc8349509040207570907od_o_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_32_mem__case__prodI2,axiom,
! [P: product_prod_o_o,Z: nat,C: $o > $o > set_nat] :
( ! [A2: $o,B2: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( member_nat @ Z @ ( C @ A2 @ B2 ) ) )
=> ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_33_mem__case__prodI,axiom,
! [Z: produc5825016348098550007at_nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Pr2645174627780777389at_nat,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ Z @ ( C @ A @ B ) )
=> ( member6956540099943067662at_nat @ Z @ ( produc4831187842317280515at_nat @ C @ ( produc760948067230524457at_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_34_mem__case__prodI,axiom,
! [Z: product_prod_o_o,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Product_prod_o_o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( member7466972457876170832od_o_o @ Z @ ( C @ A @ B ) )
=> ( member7466972457876170832od_o_o @ Z @ ( produc1858041438500342065od_o_o @ C @ ( produc760948067230524457at_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_35_mem__case__prodI,axiom,
! [Z: nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_nat,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc6938862770440504796et_nat @ C @ ( produc760948067230524457at_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_36_mem__case__prodI,axiom,
! [Z: produc5825016348098550007at_nat,C: $o > $o > set_Pr2645174627780777389at_nat,A: $o,B: $o] :
( ( member6956540099943067662at_nat @ Z @ ( C @ A @ B ) )
=> ( member6956540099943067662at_nat @ Z @ ( produc3500709797993629913at_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_37_mem__case__prodI,axiom,
! [Z: product_prod_o_o,C: $o > $o > set_Product_prod_o_o,A: $o,B: $o] :
( ( member7466972457876170832od_o_o @ Z @ ( C @ A @ B ) )
=> ( member7466972457876170832od_o_o @ Z @ ( produc8349509040207570907od_o_o @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_38_mem__case__prodI,axiom,
! [Z: nat,C: $o > $o > set_nat,A: $o,B: $o] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_39_mem__case__prodE,axiom,
! [Z: produc5825016348098550007at_nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Pr2645174627780777389at_nat,P: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Z @ ( produc4831187842317280515at_nat @ C @ P ) )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ X3 @ Y ) )
=> ~ ( member6956540099943067662at_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_40_mem__case__prodE,axiom,
! [Z: product_prod_o_o,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Product_prod_o_o,P: produc5825016348098550007at_nat] :
( ( member7466972457876170832od_o_o @ Z @ ( produc1858041438500342065od_o_o @ C @ P ) )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ X3 @ Y ) )
=> ~ ( member7466972457876170832od_o_o @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_41_mem__case__prodE,axiom,
! [Z: nat,C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_nat,P: produc5825016348098550007at_nat] :
( ( member_nat @ Z @ ( produc6938862770440504796et_nat @ C @ P ) )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ X3 @ Y ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_42_mem__case__prodE,axiom,
! [Z: produc5825016348098550007at_nat,C: $o > $o > set_Pr2645174627780777389at_nat,P: product_prod_o_o] :
( ( member6956540099943067662at_nat @ Z @ ( produc3500709797993629913at_nat @ C @ P ) )
=> ~ ! [X3: $o,Y: $o] :
( ( P
= ( product_Pair_o_o @ X3 @ Y ) )
=> ~ ( member6956540099943067662at_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_43_mem__case__prodE,axiom,
! [Z: product_prod_o_o,C: $o > $o > set_Product_prod_o_o,P: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Z @ ( produc8349509040207570907od_o_o @ C @ P ) )
=> ~ ! [X3: $o,Y: $o] :
( ( P
= ( product_Pair_o_o @ X3 @ Y ) )
=> ~ ( member7466972457876170832od_o_o @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_44_mem__case__prodE,axiom,
! [Z: nat,C: $o > $o > set_nat,P: product_prod_o_o] :
( ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ P ) )
=> ~ ! [X3: $o,Y: $o] :
( ( P
= ( product_Pair_o_o @ X3 @ Y ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_45_Pair__inject,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A5: relational_fmla_a_b,B4: set_Pr1261947904930325089at_nat] :
( ( ( produc760948067230524457at_nat @ A @ B )
= ( produc760948067230524457at_nat @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_46_Pair__inject,axiom,
! [A: $o,B: $o,A5: $o,B4: $o] :
( ( ( product_Pair_o_o @ A @ B )
= ( product_Pair_o_o @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B = (~ B4) ) ) ) ).
% Pair_inject
thf(fact_47_prod__cases,axiom,
! [P2: produc5825016348098550007at_nat > $o,P: produc5825016348098550007at_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_48_prod__cases,axiom,
! [P2: product_prod_o_o > $o,P: product_prod_o_o] :
( ! [A2: $o,B2: $o] : ( P2 @ ( product_Pair_o_o @ A2 @ B2 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_49_surj__pair,axiom,
! [P: produc5825016348098550007at_nat] :
? [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( P
= ( produc760948067230524457at_nat @ X3 @ Y ) ) ).
% surj_pair
thf(fact_50_surj__pair,axiom,
! [P: product_prod_o_o] :
? [X3: $o,Y: $o] :
( P
= ( product_Pair_o_o @ X3 @ Y ) ) ).
% surj_pair
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y3: produc5825016348098550007at_nat] :
~ ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( Y3
!= ( produc760948067230524457at_nat @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_o_o] :
~ ! [A2: $o,B2: $o] :
( Y3
!= ( product_Pair_o_o @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_53_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_54_ex__in__conv,axiom,
! [A4: set_Product_prod_o_o] :
( ( ? [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A4 ) )
= ( A4 != bot_bo7073875226086086771od_o_o ) ) ).
% ex_in_conv
thf(fact_55_ex__in__conv,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( ? [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A4 ) )
= ( A4 != bot_bo1973934853101755969at_nat ) ) ).
% ex_in_conv
thf(fact_56_equals0I,axiom,
! [A4: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_57_equals0I,axiom,
! [A4: set_Product_prod_o_o] :
( ! [Y: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ Y @ A4 )
=> ( A4 = bot_bo7073875226086086771od_o_o ) ) ).
% equals0I
thf(fact_58_equals0I,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ! [Y: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ Y @ A4 )
=> ( A4 = bot_bo1973934853101755969at_nat ) ) ).
% equals0I
thf(fact_59_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_60_equals0D,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o] :
( ( A4 = bot_bo7073875226086086771od_o_o )
=> ~ ( member7466972457876170832od_o_o @ A @ A4 ) ) ).
% equals0D
thf(fact_61_equals0D,axiom,
! [A4: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( A4 = bot_bo1973934853101755969at_nat )
=> ~ ( member6956540099943067662at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_62_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_63_emptyE,axiom,
! [A: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ).
% emptyE
thf(fact_64_emptyE,axiom,
! [A: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ A @ bot_bo1973934853101755969at_nat ) ).
% emptyE
thf(fact_65_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_66_Set_Oempty__def,axiom,
( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] : $false ) ) ).
% Set.empty_def
thf(fact_67_Set_Oempty__def,axiom,
( bot_bo1973934853101755969at_nat
= ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : $false ) ) ).
% Set.empty_def
thf(fact_68_pigeonhole__infinite__rel,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat,R: product_prod_o_o > produc5825016348098550007at_nat > $o] :
( ~ ( finite6120865539452801872od_o_o @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B5 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B5 )
& ~ ( finite6120865539452801872od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [A3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A3 @ A4 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_69_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat,R: nat > produc5825016348098550007at_nat > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B5 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A4 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_70_pigeonhole__infinite__rel,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,R: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ~ ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ B5 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ B5 )
& ~ ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [A3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A3 @ A4 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_71_not__finite__existsD,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
( ~ ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P2 ) )
=> ? [X_1: produc5825016348098550007at_nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_72_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,Prod: produc5825016348098550007at_nat] :
( ( H @ ( produc5861013030283631554_nat_o @ F @ Prod ) )
= ( produc5861013030283631554_nat_o
@ ^ [X12: relational_fmla_a_b,X22: set_Pr1261947904930325089at_nat] : ( H @ ( F @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_73_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: $o > $o > $o,Prod: product_prod_o_o] :
( ( H @ ( produc6197397395684419436_o_o_o @ F @ Prod ) )
= ( produc6197397395684419436_o_o_o
@ ^ [X12: $o,X22: $o] : ( H @ ( F @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_74_mem__Collect__eq,axiom,
! [A: product_prod_o_o,P2: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ A @ ( collec3167064739498627218od_o_o @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: produc5825016348098550007at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( member6956540099943067662at_nat @ A @ ( collec7697611494764367948at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A4: set_Product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_80_Collect__cong,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ! [X3: produc5825016348098550007at_nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7697611494764367948at_nat @ P2 )
= ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_81_prod_Odisc__eq__case,axiom,
! [Prod: produc5825016348098550007at_nat] :
( produc5861013030283631554_nat_o
@ ^ [Uu: relational_fmla_a_b,Uv: set_Pr1261947904930325089at_nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_82_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_o_o] :
( produc6197397395684419436_o_o_o
@ ^ [Uu: $o,Uv: $o] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_83_old_Oprod_Ocase,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,X1: relational_fmla_a_b,X2: set_Pr1261947904930325089at_nat] :
( ( produc5861013030283631554_nat_o @ F @ ( produc760948067230524457at_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_84_old_Oprod_Ocase,axiom,
! [F: $o > $o > $o,X1: $o,X2: $o] :
( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_85_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_86_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_o_o] :
( ~ ( finite6120865539452801872od_o_o @ S )
=> ( S != bot_bo7073875226086086771od_o_o ) ) ).
% infinite_imp_nonempty
thf(fact_87_infinite__imp__nonempty,axiom,
! [S: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ S )
=> ( S != bot_bo1973934853101755969at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_88_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_89_finite_OemptyI,axiom,
finite6120865539452801872od_o_o @ bot_bo7073875226086086771od_o_o ).
% finite.emptyI
thf(fact_90_finite_OemptyI,axiom,
finite6790245451575510286at_nat @ bot_bo1973934853101755969at_nat ).
% finite.emptyI
thf(fact_91_cond__case__prod__eta,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,G: produc5825016348098550007at_nat > $o] :
( ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( F @ X3 @ Y )
= ( G @ ( produc760948067230524457at_nat @ X3 @ Y ) ) )
=> ( ( produc5861013030283631554_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_92_cond__case__prod__eta,axiom,
! [F: $o > $o > $o,G: product_prod_o_o > $o] :
( ! [X3: $o,Y: $o] :
( ( F @ X3 @ Y )
= ( G @ ( product_Pair_o_o @ X3 @ Y ) ) )
=> ( ( produc6197397395684419436_o_o_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_93_case__prod__eta,axiom,
! [F: produc5825016348098550007at_nat > $o] :
( ( produc5861013030283631554_nat_o
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( F @ ( produc760948067230524457at_nat @ X @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_94_case__prod__eta,axiom,
! [F: product_prod_o_o > $o] :
( ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] : ( F @ ( product_Pair_o_o @ X @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_95_case__prodE2,axiom,
! [Q: $o > $o,P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,Z: produc5825016348098550007at_nat] :
( ( Q @ ( produc5861013030283631554_nat_o @ P2 @ Z ) )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( Z
= ( produc760948067230524457at_nat @ X3 @ Y ) )
=> ~ ( Q @ ( P2 @ X3 @ Y ) ) ) ) ).
% case_prodE2
thf(fact_96_case__prodE2,axiom,
! [Q: $o > $o,P2: $o > $o > $o,Z: product_prod_o_o] :
( ( Q @ ( produc6197397395684419436_o_o_o @ P2 @ Z ) )
=> ~ ! [X3: $o,Y: $o] :
( ( Z
= ( product_Pair_o_o @ X3 @ Y ) )
=> ~ ( Q @ ( P2 @ X3 @ Y ) ) ) ) ).
% case_prodE2
thf(fact_97_case__prodE,axiom,
! [C: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,P: produc5825016348098550007at_nat] :
( ( produc5861013030283631554_nat_o @ C @ P )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ X3 @ Y ) )
=> ~ ( C @ X3 @ Y ) ) ) ).
% case_prodE
thf(fact_98_case__prodE,axiom,
! [C: $o > $o > $o,P: product_prod_o_o] :
( ( produc6197397395684419436_o_o_o @ C @ P )
=> ~ ! [X3: $o,Y: $o] :
( ( P
= ( product_Pair_o_o @ X3 @ Y ) )
=> ~ ( C @ X3 @ Y ) ) ) ).
% case_prodE
thf(fact_99_case__prodD,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( produc5861013030283631554_nat_o @ F @ ( produc760948067230524457at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_100_case__prodD,axiom,
! [F: $o > $o > $o,A: $o,B: $o] :
( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_101_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z2: nat] :
( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_102_finite__transitivity__chain,axiom,
! [A4: set_Product_prod_o_o,R: product_prod_o_o > product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ! [X3: product_prod_o_o] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: product_prod_o_o,Y: product_prod_o_o,Z2: product_prod_o_o] :
( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ? [Y5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4 = bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_103_finite__transitivity__chain,axiom,
! [A4: set_Pr2645174627780777389at_nat,R: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ! [X3: produc5825016348098550007at_nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: produc5825016348098550007at_nat,Y: produc5825016348098550007at_nat,Z2: produc5825016348098550007at_nat] :
( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ? [Y5: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y5 @ A4 )
& ( R @ X3 @ Y5 ) ) )
=> ( A4 = bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_104_param__case__prod_H_H,axiom,
! [P: produc5825016348098550007at_nat,P3: produc5825016348098550007at_nat,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > relational_fmla_a_b,F2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,R: set_Pr2645174627780777389at_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,A6: relational_fmla_a_b,B6: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( ( P3
= ( produc760948067230524457at_nat @ A6 @ B6 ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( produc7155469188641419765la_a_b @ F @ P ) @ ( produc8812027116810229899at_nat @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_105_param__case__prod_H_H,axiom,
! [P: produc5825016348098550007at_nat,P3: product_prod_o_o,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > relational_fmla_a_b,F2: $o > $o > set_Pr1261947904930325089at_nat,R: set_Pr2645174627780777389at_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,A6: $o,B6: $o] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( ( P3
= ( product_Pair_o_o @ A6 @ B6 ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( produc7155469188641419765la_a_b @ F @ P ) @ ( produc2861560147193911861at_nat @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_106_param__case__prod_H_H,axiom,
! [P: product_prod_o_o,P3: produc5825016348098550007at_nat,F: $o > $o > relational_fmla_a_b,F2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,R: set_Pr2645174627780777389at_nat] :
( ! [A2: $o,B2: $o,A6: relational_fmla_a_b,B6: set_Pr1261947904930325089at_nat] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( ( P3
= ( produc760948067230524457at_nat @ A6 @ B6 ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( produc8736635878069395147la_a_b @ F @ P ) @ ( produc8812027116810229899at_nat @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_107_param__case__prod_H_H,axiom,
! [P: product_prod_o_o,P3: product_prod_o_o,F: $o > $o > relational_fmla_a_b,F2: $o > $o > set_Pr1261947904930325089at_nat,R: set_Pr2645174627780777389at_nat] :
( ! [A2: $o,B2: $o,A6: $o,B6: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( ( P3
= ( product_Pair_o_o @ A6 @ B6 ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( produc8736635878069395147la_a_b @ F @ P ) @ ( produc2861560147193911861at_nat @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_108_param__case__prod_H_H,axiom,
! [P: produc5825016348098550007at_nat,P3: produc5825016348098550007at_nat,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,F2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,R: set_Product_prod_o_o] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,A6: relational_fmla_a_b,B6: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( ( P3
= ( produc760948067230524457at_nat @ A6 @ B6 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( produc5861013030283631554_nat_o @ F @ P ) @ ( produc5861013030283631554_nat_o @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_109_param__case__prod_H_H,axiom,
! [P: produc5825016348098550007at_nat,P3: product_prod_o_o,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,F2: $o > $o > $o,R: set_Product_prod_o_o] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,A6: $o,B6: $o] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( ( P3
= ( product_Pair_o_o @ A6 @ B6 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( produc5861013030283631554_nat_o @ F @ P ) @ ( produc6197397395684419436_o_o_o @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_110_param__case__prod_H_H,axiom,
! [P: product_prod_o_o,P3: produc5825016348098550007at_nat,F: $o > $o > $o,F2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,R: set_Product_prod_o_o] :
( ! [A2: $o,B2: $o,A6: relational_fmla_a_b,B6: set_Pr1261947904930325089at_nat] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( ( P3
= ( produc760948067230524457at_nat @ A6 @ B6 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( produc6197397395684419436_o_o_o @ F @ P ) @ ( produc5861013030283631554_nat_o @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_111_param__case__prod_H_H,axiom,
! [P: product_prod_o_o,P3: product_prod_o_o,F: $o > $o > $o,F2: $o > $o > $o,R: set_Product_prod_o_o] :
( ! [A2: $o,B2: $o,A6: $o,B6: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( ( P3
= ( product_Pair_o_o @ A6 @ B6 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 @ B2 ) @ ( F2 @ A6 @ B6 ) ) @ R ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( produc6197397395684419436_o_o_o @ F @ P ) @ ( produc6197397395684419436_o_o_o @ F2 @ P3 ) ) @ R ) ) ).
% param_case_prod''
thf(fact_112_split__cong,axiom,
! [Q2: produc5825016348098550007at_nat,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,G: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,P: produc5825016348098550007at_nat] :
( ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( ( produc760948067230524457at_nat @ X3 @ Y )
= Q2 )
=> ( ( F @ X3 @ Y )
= ( G @ X3 @ Y ) ) )
=> ( ( P = Q2 )
=> ( ( produc5861013030283631554_nat_o @ F @ P )
= ( produc5861013030283631554_nat_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_113_split__cong,axiom,
! [Q2: product_prod_o_o,F: $o > $o > $o,G: $o > $o > $o,P: product_prod_o_o] :
( ! [X3: $o,Y: $o] :
( ( ( product_Pair_o_o @ X3 @ Y )
= Q2 )
=> ( ( F @ X3 @ Y )
= ( G @ X3 @ Y ) ) )
=> ( ( P = Q2 )
=> ( ( produc6197397395684419436_o_o_o @ F @ P )
= ( produc6197397395684419436_o_o_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_114_bot__prod__def,axiom,
( bot_bo6949252958386473875od_o_o
= ( product_Pair_o_o @ bot_bot_o @ bot_bot_o ) ) ).
% bot_prod_def
thf(fact_115_bot__prod__def,axiom,
( bot_bo3047382831089536473et_nat
= ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_116_bot__prod__def,axiom,
( bot_bo1661953738785618570od_o_o
= ( produc2230572554435680680od_o_o @ bot_bot_set_nat @ bot_bo7073875226086086771od_o_o ) ) ).
% bot_prod_def
thf(fact_117_bot__prod__def,axiom,
( bot_bo4057417303566359680at_nat
= ( produc7884731453098325580at_nat @ bot_bot_set_nat @ bot_bo1973934853101755969at_nat ) ) ).
% bot_prod_def
thf(fact_118_bot__prod__def,axiom,
( bot_bo7523401752186168528et_nat
= ( produc3326108477681520182et_nat @ bot_bo7073875226086086771od_o_o @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_119_bot__prod__def,axiom,
( bot_bo6876824833658055507od_o_o
= ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) ) ).
% bot_prod_def
thf(fact_120_bot__prod__def,axiom,
( bot_bo4428949417863240183at_nat
= ( produc8468126894364568669at_nat @ bot_bo7073875226086086771od_o_o @ bot_bo1973934853101755969at_nat ) ) ).
% bot_prod_def
thf(fact_121_bot__prod__def,axiom,
( bot_bo4644590366274264960et_nat
= ( produc7713712508742389708et_nat @ bot_bo1973934853101755969at_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_122_bot__prod__def,axiom,
( bot_bo5107019627512315043od_o_o
= ( produc3930545150249737025od_o_o @ bot_bo1973934853101755969at_nat @ bot_bo7073875226086086771od_o_o ) ) ).
% bot_prod_def
thf(fact_123_bot__prod__def,axiom,
( bot_bo1315690980769176743at_nat
= ( produc4461856759730394867at_nat @ bot_bo1973934853101755969at_nat @ bot_bo1973934853101755969at_nat ) ) ).
% bot_prod_def
thf(fact_124_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A7: set_nat] : ( A7 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_125_Set_Ois__empty__def,axiom,
( is_emp697939970857919993od_o_o
= ( ^ [A7: set_Product_prod_o_o] : ( A7 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Set.is_empty_def
thf(fact_126_Set_Ois__empty__def,axiom,
( is_emp7968975513341040805at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat] : ( A7 = bot_bo1973934853101755969at_nat ) ) ) ).
% Set.is_empty_def
thf(fact_127_internal__case__prod__def,axiom,
produc7512522816056521368_nat_o = produc5861013030283631554_nat_o ).
% internal_case_prod_def
thf(fact_128_internal__case__prod__def,axiom,
produc331225583472661270_o_o_o = produc6197397395684419436_o_o_o ).
% internal_case_prod_def
thf(fact_129_case__prod__Pair__iden,axiom,
! [P: produc5825016348098550007at_nat] :
( ( produc3455528468001603661at_nat @ produc760948067230524457at_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_130_case__prod__Pair__iden,axiom,
! [P: product_prod_o_o] :
( ( produc7436348682273225467od_o_o @ product_Pair_o_o @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_131_pred__equals__eq2,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( ( ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ R ) )
= ( ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_132_pred__equals__eq2,axiom,
! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R ) )
= ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_133_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_134_bot__empty__eq,axiom,
( bot_bo6429049293326997034_o_o_o
= ( ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ bot_bo7073875226086086771od_o_o ) ) ) ).
% bot_empty_eq
thf(fact_135_bot__empty__eq,axiom,
( bot_bo3884824731534426500_nat_o
= ( ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ bot_bo1973934853101755969at_nat ) ) ) ).
% bot_empty_eq
thf(fact_136_bot__empty__eq2,axiom,
( bot_bot_o_o_o
= ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ bot_bo7073875226086086771od_o_o ) ) ) ).
% bot_empty_eq2
thf(fact_137_bot__empty__eq2,axiom,
( bot_bo7672613165809151790_nat_o
= ( ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ bot_bo1973934853101755969at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_138_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_139_bot__set__def,axiom,
( bot_bo7073875226086086771od_o_o
= ( collec3167064739498627218od_o_o @ bot_bo6429049293326997034_o_o_o ) ) ).
% bot_set_def
thf(fact_140_bot__set__def,axiom,
( bot_bo1973934853101755969at_nat
= ( collec7697611494764367948at_nat @ bot_bo3884824731534426500_nat_o ) ) ).
% bot_set_def
thf(fact_141_bex2I,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,S: set_Pr2645174627780777389at_nat,P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ S )
=> ( ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ S )
=> ( P2 @ A @ B ) )
=> ? [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A2 @ B2 ) @ S )
& ( P2 @ A2 @ B2 ) ) ) ) ).
% bex2I
thf(fact_142_bex2I,axiom,
! [A: $o,B: $o,S: set_Product_prod_o_o,P2: $o > $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ S )
=> ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ S )
=> ( P2 @ A @ B ) )
=> ? [A2: $o,B2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ S )
& ( P2 @ A2 @ B2 ) ) ) ) ).
% bex2I
thf(fact_143_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_144_set__notEmptyE,axiom,
! [S: set_Product_prod_o_o] :
( ( S != bot_bo7073875226086086771od_o_o )
=> ~ ! [X3: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_145_set__notEmptyE,axiom,
! [S: set_Pr2645174627780777389at_nat] :
( ( S != bot_bo1973934853101755969at_nat )
=> ~ ! [X3: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_146_memb__imp__not__empty,axiom,
! [X4: nat,S: set_nat] :
( ( member_nat @ X4 @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_147_memb__imp__not__empty,axiom,
! [X4: product_prod_o_o,S: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ S )
=> ( S != bot_bo7073875226086086771od_o_o ) ) ).
% memb_imp_not_empty
thf(fact_148_memb__imp__not__empty,axiom,
! [X4: produc5825016348098550007at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X4 @ S )
=> ( S != bot_bo1973934853101755969at_nat ) ) ).
% memb_imp_not_empty
thf(fact_149_inv__image__def,axiom,
( inv_image_o_o
= ( ^ [R2: set_Product_prod_o_o,F3: $o > $o] :
( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F3 @ X ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
thf(fact_150_uncurry__def,axiom,
uncurr1240319265416644269_nat_o = produc5861013030283631554_nat_o ).
% uncurry_def
thf(fact_151_uncurry__def,axiom,
uncurry_o_o_o = produc6197397395684419436_o_o_o ).
% uncurry_def
thf(fact_152_finite__SigmaI2,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo3083307316010499117at_nat ) ) ) )
=> ( ! [A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A4 )
=> ( finite9047747110432174090at_nat @ ( B5 @ A2 ) ) )
=> ( finite6790245451575510286at_nat @ ( produc7082850253950630328at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_153_finite__SigmaI2,axiom,
! [A4: set_Product_prod_o_o,B5: product_prod_o_o > set_nat] :
( ( finite6120865539452801872od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
& ( ( B5 @ X )
!= bot_bot_set_nat ) ) ) )
=> ( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A2 ) ) )
=> ( finite2044766826296809277_o_nat @ ( produc2027885580938973615_o_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_154_finite__SigmaI2,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bot_set_nat ) ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A2 ) ) )
=> ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_155_finite__SigmaI2,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_nat] :
( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bot_set_nat ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A2 ) ) )
=> ( finite1428100398092564177at_nat @ ( produc3607307171264959249at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_156_finite__SigmaI2,axiom,
! [A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo7073875226086086771od_o_o ) ) ) )
=> ( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( finite6120865539452801872od_o_o @ ( B5 @ A2 ) ) )
=> ( finite5612094862846137872od_o_o @ ( produc6514336722952383752od_o_o @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_157_finite__SigmaI2,axiom,
! [A4: set_nat,B5: nat > set_Product_prod_o_o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo7073875226086086771od_o_o ) ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( finite6120865539452801872od_o_o @ ( B5 @ A2 ) ) )
=> ( finite322770798450611171od_o_o @ ( produc7535496041681465101od_o_o @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_158_finite__SigmaI2,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Product_prod_o_o] :
( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo7073875226086086771od_o_o ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( finite6120865539452801872od_o_o @ ( B5 @ A2 ) ) )
=> ( finite688782994846339964od_o_o @ ( produc7724249792059183142od_o_o @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_159_finite__SigmaI2,axiom,
! [A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( finite6120865539452801872od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo1973934853101755969at_nat ) ) ) )
=> ( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite9006603388738057188at_nat @ ( produc8851762376515582294at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_160_finite__SigmaI2,axiom,
! [A4: set_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo1973934853101755969at_nat ) ) ) )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite3666854805591244497at_nat @ ( produc5885360257917011089at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_161_finite__SigmaI2,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ( ( B5 @ X )
!= bot_bo1973934853101755969at_nat ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite8185985059073859960at_nat @ ( produc984725824077784504at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_162_finite__cartesian__productD1,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( finite6790245451575510286at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) )
=> ( ( B5 != bot_bo3083307316010499117at_nat )
=> ( finite5600759454172676150la_a_b @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_163_finite__cartesian__productD1,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat] :
( ( finite1428100398092564177at_nat
@ ( produc3607307171264959249at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_164_finite__cartesian__productD1,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o] :
( ( finite688782994846339964od_o_o
@ ( produc7724249792059183142od_o_o @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
=> ( ( B5 != bot_bo7073875226086086771od_o_o )
=> ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_165_finite__cartesian__productD1,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite8185985059073859960at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
=> ( ( B5 != bot_bo1973934853101755969at_nat )
=> ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% finite_cartesian_productD1
thf(fact_166_finite__cartesian__productD2,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( finite6790245451575510286at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) )
=> ( ( A4 != bot_bo4495933725496725865la_a_b )
=> ( finite9047747110432174090at_nat @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_167_finite__cartesian__productD2,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite3666854805591244497at_nat
@ ( produc5885360257917011089at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( finite6790245451575510286at_nat @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_168_finite__cartesian__productD2,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat] :
( ( finite9006603388738057188at_nat
@ ( produc8851762376515582294at_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 ) )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( finite6790245451575510286at_nat @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_169_finite__cartesian__productD2,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite8185985059073859960at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
=> ( ( A4 != bot_bo1973934853101755969at_nat )
=> ( finite6790245451575510286at_nat @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_170_finite__cartesian__product__iff,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( finite6790245451575510286at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) )
= ( ( A4 = bot_bo4495933725496725865la_a_b )
| ( B5 = bot_bo3083307316010499117at_nat )
| ( ( finite5600759454172676150la_a_b @ A4 )
& ( finite9047747110432174090at_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_171_finite__cartesian__product__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite6177210948735845034at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat )
| ( ( finite_finite_nat @ A4 )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_172_finite__cartesian__product__iff,axiom,
! [A4: set_nat,B5: set_Product_prod_o_o] :
( ( finite322770798450611171od_o_o
@ ( produc7535496041681465101od_o_o @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo7073875226086086771od_o_o )
| ( ( finite_finite_nat @ A4 )
& ( finite6120865539452801872od_o_o @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_173_finite__cartesian__product__iff,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite3666854805591244497at_nat
@ ( produc5885360257917011089at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo1973934853101755969at_nat )
| ( ( finite_finite_nat @ A4 )
& ( finite6790245451575510286at_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_174_finite__cartesian__product__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_nat] :
( ( finite2044766826296809277_o_nat
@ ( produc2027885580938973615_o_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bot_set_nat )
| ( ( finite6120865539452801872od_o_o @ A4 )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_175_finite__cartesian__product__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( finite5612094862846137872od_o_o
@ ( produc6514336722952383752od_o_o @ A4
@ ^ [Uu: product_prod_o_o] : B5 ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo7073875226086086771od_o_o )
| ( ( finite6120865539452801872od_o_o @ A4 )
& ( finite6120865539452801872od_o_o @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_176_finite__cartesian__product__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat] :
( ( finite9006603388738057188at_nat
@ ( produc8851762376515582294at_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo1973934853101755969at_nat )
| ( ( finite6120865539452801872od_o_o @ A4 )
& ( finite6790245451575510286at_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_177_finite__cartesian__product__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat] :
( ( finite1428100398092564177at_nat
@ ( produc3607307171264959249at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bot_set_nat )
| ( ( finite6790245451575510286at_nat @ A4 )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_178_finite__cartesian__product__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o] :
( ( finite688782994846339964od_o_o
@ ( produc7724249792059183142od_o_o @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bo7073875226086086771od_o_o )
| ( ( finite6790245451575510286at_nat @ A4 )
& ( finite6120865539452801872od_o_o @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_179_finite__cartesian__product__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite8185985059073859960at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bo1973934853101755969at_nat )
| ( ( finite6790245451575510286at_nat @ A4 )
& ( finite6790245451575510286at_nat @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_180_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_181_Collect__empty__eq__bot,axiom,
! [P2: product_prod_o_o > $o] :
( ( ( collec3167064739498627218od_o_o @ P2 )
= bot_bo7073875226086086771od_o_o )
= ( P2 = bot_bo6429049293326997034_o_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_182_Collect__empty__eq__bot,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
( ( ( collec7697611494764367948at_nat @ P2 )
= bot_bo1973934853101755969at_nat )
= ( P2 = bot_bo3884824731534426500_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_183_mem__Sigma__iff,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_184_mem__Sigma__iff,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
= ( ( member_o @ A @ A4 )
& ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_185_mem__Sigma__iff,axiom,
! [A: product_prod_o_o,B: nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_nat] :
( ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ ( produc2027885580938973615_o_nat @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_186_mem__Sigma__iff,axiom,
! [A: nat,B: product_prod_o_o,A4: set_nat,B5: nat > set_Product_prod_o_o] :
( ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ ( produc7535496041681465101od_o_o @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_187_mem__Sigma__iff,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( produc6514336722952383752od_o_o @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ A @ A4 )
& ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_188_mem__Sigma__iff,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc7082850253950630328at_nat @ A4 @ B5 ) )
= ( ( member4680049679412964150la_a_b @ A @ A4 )
& ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_189_mem__Sigma__iff,axiom,
! [A: produc5825016348098550007at_nat,B: nat,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_nat] :
( ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ ( produc3607307171264959249at_nat @ A4 @ B5 ) )
= ( ( member6956540099943067662at_nat @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_190_mem__Sigma__iff,axiom,
! [A: nat,B: produc5825016348098550007at_nat,A4: set_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ ( produc5885360257917011089at_nat @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member6956540099943067662at_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_191_mem__Sigma__iff,axiom,
! [A: produc5825016348098550007at_nat,B: product_prod_o_o,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Product_prod_o_o] :
( ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ ( produc7724249792059183142od_o_o @ A4 @ B5 ) )
= ( ( member6956540099943067662at_nat @ A @ A4 )
& ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_192_mem__Sigma__iff,axiom,
! [A: product_prod_o_o,B: produc5825016348098550007at_nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ ( produc8851762376515582294at_nat @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ A @ A4 )
& ( member6956540099943067662at_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_193_SigmaI,axiom,
! [A: nat,A4: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_194_SigmaI,axiom,
! [A: $o,A4: set_o,B: $o,B5: $o > set_o] :
( ( member_o @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_195_SigmaI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: nat,B5: product_prod_o_o > set_nat] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ ( produc2027885580938973615_o_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_196_SigmaI,axiom,
! [A: nat,A4: set_nat,B: product_prod_o_o,B5: nat > set_Product_prod_o_o] :
( ( member_nat @ A @ A4 )
=> ( ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) )
=> ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ ( produc7535496041681465101od_o_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_197_SigmaI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: product_prod_o_o,B5: product_prod_o_o > set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( produc6514336722952383752od_o_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_198_SigmaI,axiom,
! [A: relational_fmla_a_b,A4: set_Re381260168593705685la_a_b,B: set_Pr1261947904930325089at_nat,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member4680049679412964150la_a_b @ A @ A4 )
=> ( ( member2643936169264416010at_nat @ B @ ( B5 @ A ) )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc7082850253950630328at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_199_SigmaI,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: nat,B5: produc5825016348098550007at_nat > set_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ ( produc3607307171264959249at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_200_SigmaI,axiom,
! [A: nat,A4: set_nat,B: produc5825016348098550007at_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member6956540099943067662at_nat @ B @ ( B5 @ A ) )
=> ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ ( produc5885360257917011089at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_201_SigmaI,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: product_prod_o_o,B5: produc5825016348098550007at_nat > set_Product_prod_o_o] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) )
=> ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ ( produc7724249792059183142od_o_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_202_SigmaI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: produc5825016348098550007at_nat,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member6956540099943067662at_nat @ B @ ( B5 @ A ) )
=> ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ ( produc8851762376515582294at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_203_in__inv__image,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,F: $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( inv_image_o_o @ R3 @ F ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ X4 ) @ ( F @ Y3 ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_204_Collect__case__prod,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ^ [A3: produc5825016348098550007at_nat,B3: produc5825016348098550007at_nat] :
( ( P2 @ A3 )
& ( Q @ B3 ) ) ) )
= ( produc984725824077784504at_nat @ ( collec7697611494764367948at_nat @ P2 )
@ ^ [Uu: produc5825016348098550007at_nat] : ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_205_Collect__case__prod,axiom,
! [P2: relational_fmla_a_b > $o,Q: set_Pr1261947904930325089at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat] :
( ( P2 @ A3 )
& ( Q @ B3 ) ) ) )
= ( produc7082850253950630328at_nat @ ( collec3419995626248312948la_a_b @ P2 )
@ ^ [Uu: relational_fmla_a_b] : ( collec5514110066124741708at_nat @ Q ) ) ) ).
% Collect_case_prod
thf(fact_206_Collect__case__prod,axiom,
! [P2: $o > $o,Q: $o > $o] :
( ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] :
( ( P2 @ A3 )
& ( Q @ B3 ) ) ) )
= ( product_Sigma_o_o @ ( collect_o @ P2 )
@ ^ [Uu: $o] : ( collect_o @ Q ) ) ) ).
% Collect_case_prod
thf(fact_207_Sigma__empty1,axiom,
! [B5: $o > set_o] :
( ( product_Sigma_o_o @ bot_bot_set_o @ B5 )
= bot_bo7073875226086086771od_o_o ) ).
% Sigma_empty1
thf(fact_208_Sigma__empty1,axiom,
! [B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( produc7082850253950630328at_nat @ bot_bo4495933725496725865la_a_b @ B5 )
= bot_bo1973934853101755969at_nat ) ).
% Sigma_empty1
thf(fact_209_Sigma__empty2,axiom,
! [A4: set_o] :
( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Sigma_empty2
thf(fact_210_Sigma__empty2,axiom,
! [A4: set_Re381260168593705685la_a_b] :
( ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : bot_bo3083307316010499117at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Sigma_empty2
thf(fact_211_Times__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo2099793752762293965at_nat )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_212_Times__empty,axiom,
! [A4: set_o,B5: set_o] :
( ( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo7073875226086086771od_o_o )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_213_Times__empty,axiom,
! [A4: set_nat,B5: set_Product_prod_o_o] :
( ( ( produc7535496041681465101od_o_o @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo3519519488074456726od_o_o )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Times_empty
thf(fact_214_Times__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_nat] :
( ( ( produc2027885580938973615_o_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= bot_bo6315691432038765808_o_nat )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_215_Times__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( produc6514336722952383752od_o_o @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= bot_bo2881837355931550515od_o_o )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Times_empty
thf(fact_216_Times__empty,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 )
= bot_bo1973934853101755969at_nat )
= ( ( A4 = bot_bo4495933725496725865la_a_b )
| ( B5 = bot_bo3083307316010499117at_nat ) ) ) ).
% Times_empty
thf(fact_217_Times__empty,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( produc5885360257917011089at_nat @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo2461228429094800244at_nat )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo1973934853101755969at_nat ) ) ) ).
% Times_empty
thf(fact_218_Times__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat] :
( ( ( produc3607307171264959249at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 )
= bot_bo7988639415687791476at_nat )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_219_Times__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat] :
( ( ( produc8851762376515582294at_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= bot_bo8958214773085927959at_nat )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo1973934853101755969at_nat ) ) ) ).
% Times_empty
thf(fact_220_Times__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o] :
( ( ( produc7724249792059183142od_o_o @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 )
= bot_bo1775769411710406063od_o_o )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Times_empty
thf(fact_221_finite__SigmaI,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( finite5600759454172676150la_a_b @ A4 )
=> ( ! [A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A4 )
=> ( finite9047747110432174090at_nat @ ( B5 @ A2 ) ) )
=> ( finite6790245451575510286at_nat @ ( produc7082850253950630328at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI
thf(fact_222_finite__SigmaI,axiom,
! [A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite9006603388738057188at_nat @ ( produc8851762376515582294at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI
thf(fact_223_finite__SigmaI,axiom,
! [A4: set_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite3666854805591244497at_nat @ ( produc5885360257917011089at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI
thf(fact_224_finite__SigmaI,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ! [A2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( finite6790245451575510286at_nat @ ( B5 @ A2 ) ) )
=> ( finite8185985059073859960at_nat @ ( produc984725824077784504at_nat @ A4 @ B5 ) ) ) ) ).
% finite_SigmaI
thf(fact_225_SigmaE2,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_226_SigmaE2,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ~ ( ( member_o @ A @ A4 )
=> ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_227_SigmaE2,axiom,
! [A: product_prod_o_o,B: nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_nat] :
( ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ ( produc2027885580938973615_o_nat @ A4 @ B5 ) )
=> ~ ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_228_SigmaE2,axiom,
! [A: nat,B: product_prod_o_o,A4: set_nat,B5: nat > set_Product_prod_o_o] :
( ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ ( produc7535496041681465101od_o_o @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_229_SigmaE2,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( produc6514336722952383752od_o_o @ A4 @ B5 ) )
=> ~ ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ~ ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_230_SigmaE2,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc7082850253950630328at_nat @ A4 @ B5 ) )
=> ~ ( ( member4680049679412964150la_a_b @ A @ A4 )
=> ~ ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_231_SigmaE2,axiom,
! [A: produc5825016348098550007at_nat,B: nat,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_nat] :
( ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ ( produc3607307171264959249at_nat @ A4 @ B5 ) )
=> ~ ( ( member6956540099943067662at_nat @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_232_SigmaE2,axiom,
! [A: nat,B: produc5825016348098550007at_nat,A4: set_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ ( produc5885360257917011089at_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member6956540099943067662at_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_233_SigmaE2,axiom,
! [A: produc5825016348098550007at_nat,B: product_prod_o_o,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Product_prod_o_o] :
( ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ ( produc7724249792059183142od_o_o @ A4 @ B5 ) )
=> ~ ( ( member6956540099943067662at_nat @ A @ A4 )
=> ~ ( member7466972457876170832od_o_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_234_SigmaE2,axiom,
! [A: product_prod_o_o,B: produc5825016348098550007at_nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ ( produc8851762376515582294at_nat @ A4 @ B5 ) )
=> ~ ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ~ ( member6956540099943067662at_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_235_SigmaD2,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc7082850253950630328at_nat @ A4 @ B5 ) )
=> ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_236_SigmaD2,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ( member_o @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_237_SigmaD1,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc7082850253950630328at_nat @ A4 @ B5 ) )
=> ( member4680049679412964150la_a_b @ A @ A4 ) ) ).
% SigmaD1
thf(fact_238_SigmaD1,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ( member_o @ A @ A4 ) ) ).
% SigmaD1
thf(fact_239_SigmaE,axiom,
! [C: product_prod_nat_nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_240_SigmaE,axiom,
! [C: product_prod_o_o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( product_Pair_o_o @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_241_SigmaE,axiom,
! [C: produc1721968933332707494_o_nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_nat] :
( ( member5772852833339327037_o_nat @ C @ ( produc2027885580938973615_o_nat @ A4 @ B5 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc5642905467325023648_o_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_242_SigmaE,axiom,
! [C: produc9223344942341285196od_o_o,A4: set_nat,B5: nat > set_Product_prod_o_o] :
( ( member4050856805493128931od_o_o @ C @ ( produc7535496041681465101od_o_o @ A4 @ B5 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc1927143891212739326od_o_o @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_243_SigmaE,axiom,
! [C: produc4465838814804890855od_o_o,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ C @ ( produc6514336722952383752od_o_o @ A4 @ B5 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ! [Y: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc3331439552374987607od_o_o @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_244_SigmaE,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Re381260168593705685la_a_b,B5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ C @ ( produc7082850253950630328at_nat @ A4 @ B5 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A4 )
=> ! [Y: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc760948067230524457at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_245_SigmaE,axiom,
! [C: produc3676383722749475752at_nat,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_nat] :
( ( member6199062629724834769at_nat @ C @ ( produc3607307171264959249at_nat @ A4 @ B5 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc925257419237160800at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_246_SigmaE,axiom,
! [C: produc5915138130248156072at_nat,A4: set_nat,B5: nat > set_Pr2645174627780777389at_nat] :
( ( member8437817037223515089at_nat @ C @ ( produc5885360257917011089at_nat @ A4 @ B5 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc3203310505889212640at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_247_SigmaE,axiom,
! [C: produc3531288590874529381od_o_o,A4: set_Pr2645174627780777389at_nat,B5: produc5825016348098550007at_nat > set_Product_prod_o_o] :
( ( member5724791357406019196od_o_o @ C @ ( produc7724249792059183142od_o_o @ A4 @ B5 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ! [Y: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc7046591135137782167od_o_o @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_248_SigmaE,axiom,
! [C: produc2625736947911470797at_nat,A4: set_Product_prod_o_o,B5: product_prod_o_o > set_Pr2645174627780777389at_nat] :
( ( member4819239714442960612at_nat @ C @ ( produc8851762376515582294at_nat @ A4 @ B5 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ! [Y: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y @ ( B5 @ X3 ) )
=> ( C
!= ( produc8174103719594181319at_nat @ X3 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_249_Sigma__empty__iff,axiom,
! [I: set_o,X5: $o > set_o] :
( ( ( product_Sigma_o_o @ I @ X5 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( X5 @ X )
= bot_bot_set_o ) ) ) ) ).
% Sigma_empty_iff
thf(fact_250_Sigma__empty__iff,axiom,
! [I: set_Re381260168593705685la_a_b,X5: relational_fmla_a_b > set_se7855581050983116737at_nat] :
( ( ( produc7082850253950630328at_nat @ I @ X5 )
= bot_bo1973934853101755969at_nat )
= ( ! [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ I )
=> ( ( X5 @ X )
= bot_bo3083307316010499117at_nat ) ) ) ) ).
% Sigma_empty_iff
thf(fact_251_times__eq__iff,axiom,
! [A4: set_nat,B5: set_nat,C2: set_nat,D: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= ( produc457027306803732586at_nat @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_252_times__eq__iff,axiom,
! [A4: set_nat,B5: set_Product_prod_o_o,C2: set_nat,D: set_Product_prod_o_o] :
( ( ( produc7535496041681465101od_o_o @ A4
@ ^ [Uu: nat] : B5 )
= ( produc7535496041681465101od_o_o @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo7073875226086086771od_o_o ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_253_times__eq__iff,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat,C2: set_nat,D: set_Pr2645174627780777389at_nat] :
( ( ( produc5885360257917011089at_nat @ A4
@ ^ [Uu: nat] : B5 )
= ( produc5885360257917011089at_nat @ C2
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo1973934853101755969at_nat ) )
& ( ( C2 = bot_bot_set_nat )
| ( D = bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_254_times__eq__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_nat,C2: set_Product_prod_o_o,D: set_nat] :
( ( ( produc2027885580938973615_o_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= ( produc2027885580938973615_o_nat @ C2
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bot_set_nat ) )
& ( ( C2 = bot_bo7073875226086086771od_o_o )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_255_times__eq__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,C2: set_Product_prod_o_o,D: set_Product_prod_o_o] :
( ( ( produc6514336722952383752od_o_o @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= ( produc6514336722952383752od_o_o @ C2
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo7073875226086086771od_o_o ) )
& ( ( C2 = bot_bo7073875226086086771od_o_o )
| ( D = bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_256_times__eq__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat,C2: set_Product_prod_o_o,D: set_Pr2645174627780777389at_nat] :
( ( ( produc8851762376515582294at_nat @ A4
@ ^ [Uu: product_prod_o_o] : B5 )
= ( produc8851762376515582294at_nat @ C2
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( B5 = bot_bo1973934853101755969at_nat ) )
& ( ( C2 = bot_bo7073875226086086771od_o_o )
| ( D = bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_257_times__eq__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat,C2: set_Pr2645174627780777389at_nat,D: set_nat] :
( ( ( produc3607307171264959249at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 )
= ( produc3607307171264959249at_nat @ C2
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bot_set_nat ) )
& ( ( C2 = bot_bo1973934853101755969at_nat )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_258_times__eq__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o,C2: set_Pr2645174627780777389at_nat,D: set_Product_prod_o_o] :
( ( ( produc7724249792059183142od_o_o @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 )
= ( produc7724249792059183142od_o_o @ C2
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bo7073875226086086771od_o_o ) )
& ( ( C2 = bot_bo1973934853101755969at_nat )
| ( D = bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_259_times__eq__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,C2: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat] :
( ( ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 )
= ( produc984725824077784504at_nat @ C2
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( ( A4 = C2 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo1973934853101755969at_nat )
| ( B5 = bot_bo1973934853101755969at_nat ) )
& ( ( C2 = bot_bo1973934853101755969at_nat )
| ( D = bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_260_Collect__case__prod__Sigma,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ^ [X: produc5825016348098550007at_nat,Y4: produc5825016348098550007at_nat] :
( ( P2 @ X )
& ( Q @ X @ Y4 ) ) ) )
= ( produc984725824077784504at_nat @ ( collec7697611494764367948at_nat @ P2 )
@ ^ [X: produc5825016348098550007at_nat] : ( collec7697611494764367948at_nat @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_261_Collect__case__prod__Sigma,axiom,
! [P2: relational_fmla_a_b > $o,Q: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] :
( ( P2 @ X )
& ( Q @ X @ Y4 ) ) ) )
= ( produc7082850253950630328at_nat @ ( collec3419995626248312948la_a_b @ P2 )
@ ^ [X: relational_fmla_a_b] : ( collec5514110066124741708at_nat @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_262_Collect__case__prod__Sigma,axiom,
! [P2: $o > $o,Q: $o > $o > $o] :
( ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( P2 @ X )
& ( Q @ X @ Y4 ) ) ) )
= ( product_Sigma_o_o @ ( collect_o @ P2 )
@ ^ [X: $o] : ( collect_o @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_263_finite__cartesian__product,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( finite5600759454172676150la_a_b @ A4 )
=> ( ( finite9047747110432174090at_nat @ B5 )
=> ( finite6790245451575510286at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) ) ) ) ).
% finite_cartesian_product
thf(fact_264_finite__cartesian__product,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( finite8185985059073859960at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) ) ) ) ).
% finite_cartesian_product
thf(fact_265_rec__prod__is__case,axiom,
produc6824345251496796701_nat_o = produc5861013030283631554_nat_o ).
% rec_prod_is_case
thf(fact_266_rec__prod__is__case,axiom,
produc395906186863863377_o_o_o = produc6197397395684419436_o_o_o ).
% rec_prod_is_case
thf(fact_267_infinite__cartesian__product,axiom,
! [A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ~ ( finite5600759454172676150la_a_b @ A4 )
=> ( ~ ( finite9047747110432174090at_nat @ B5 )
=> ~ ( finite6790245451575510286at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_268_infinite__cartesian__product,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ A4 )
=> ( ~ ( finite6790245451575510286at_nat @ B5 )
=> ~ ( finite8185985059073859960at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : B5 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_269_member__product,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Re381260168593705685la_a_b,B5: set_se7855581050983116737at_nat] :
( ( member6956540099943067662at_nat @ X4 @ ( produc4589273740939737902at_nat @ A4 @ B5 ) )
= ( member6956540099943067662at_nat @ X4
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : B5 ) ) ) ).
% member_product
thf(fact_270_member__product,axiom,
! [X4: product_prod_o_o,A4: set_o,B5: set_o] :
( ( member7466972457876170832od_o_o @ X4 @ ( product_product_o_o @ A4 @ B5 ) )
= ( member7466972457876170832od_o_o @ X4
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) ) ) ).
% member_product
thf(fact_271_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_272_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Id_on_empty
thf(fact_273_Id__on__empty,axiom,
( ( id_on_8897472890386745284od_o_o @ bot_bo7073875226086086771od_o_o )
= bot_bo2881837355931550515od_o_o ) ).
% Id_on_empty
thf(fact_274_Id__on__empty,axiom,
( ( id_on_89751120774077850at_nat @ bot_bo1973934853101755969at_nat )
= bot_bo2147731989722242715at_nat ) ).
% Id_on_empty
thf(fact_275_bijective__Empty,axiom,
bijective_o_o @ bot_bo7073875226086086771od_o_o ).
% bijective_Empty
thf(fact_276_bijective__Empty,axiom,
biject6085764940634361907at_nat @ bot_bo1973934853101755969at_nat ).
% bijective_Empty
thf(fact_277_relation__of__def,axiom,
( order_relation_of_o
= ( ^ [P4: $o > $o > $o,A7: set_o] :
( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 )
@ ( product_Sigma_o_o @ A7
@ ^ [Uu: $o] : A7 ) )
& ( P4 @ A3 @ B3 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_278_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat] : P2 ) )
= top_to4412277702576798813at_nat ) )
& ( ~ P2
=> ( ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat] : P2 ) )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_const_case_prod
thf(fact_279_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] : P2 ) )
= top_to7721136755696657239od_o_o ) )
& ( ~ P2
=> ( ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] : P2 ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_const_case_prod
thf(fact_280_curryI,axiom,
! [F: produc5825016348098550007at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( F @ ( produc760948067230524457at_nat @ A @ B ) )
=> ( produc4986730048175820513_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_281_curryI,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( F @ ( product_Pair_o_o @ A @ B ) )
=> ( product_curry_o_o_o @ F @ A @ B ) ) ).
% curryI
thf(fact_282_UNIV__I,axiom,
! [X4: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X4 @ top_to4412277702576798813at_nat ) ).
% UNIV_I
thf(fact_283_UNIV__I,axiom,
! [X4: product_prod_o_o] : ( member7466972457876170832od_o_o @ X4 @ top_to7721136755696657239od_o_o ) ).
% UNIV_I
thf(fact_284_UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_285_finite__Plus__UNIV__iff,axiom,
( ( finite2292667041910257892at_nat @ top_to4282849876503721267at_nat )
= ( ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
& ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_286_case__prod__curry,axiom,
! [F: produc5825016348098550007at_nat > $o] :
( ( produc5861013030283631554_nat_o @ ( produc4986730048175820513_nat_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_287_case__prod__curry,axiom,
! [F: product_prod_o_o > $o] :
( ( produc6197397395684419436_o_o_o @ ( product_curry_o_o_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_288_curry__case__prod,axiom,
! [F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( produc4986730048175820513_nat_o @ ( produc5861013030283631554_nat_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_289_curry__case__prod,axiom,
! [F: $o > $o > $o] :
( ( product_curry_o_o_o @ ( produc6197397395684419436_o_o_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_290_Id__onI,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ A ) @ ( id_on_89751120774077850at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_291_Id__onI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ A ) @ ( id_on_8897472890386745284od_o_o @ A4 ) ) ) ).
% Id_onI
thf(fact_292_Id__onI,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_293_Id__onI,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( id_on_o @ A4 ) ) ) ).
% Id_onI
thf(fact_294_Collect__const,axiom,
! [P2: $o] :
( ( P2
=> ( ( collect_nat
@ ^ [S2: nat] : P2 )
= top_top_set_nat ) )
& ( ~ P2
=> ( ( collect_nat
@ ^ [S2: nat] : P2 )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_295_Collect__const,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec3167064739498627218od_o_o
@ ^ [S2: product_prod_o_o] : P2 )
= top_to7721136755696657239od_o_o ) )
& ( ~ P2
=> ( ( collec3167064739498627218od_o_o
@ ^ [S2: product_prod_o_o] : P2 )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_const
thf(fact_296_Collect__const,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec7697611494764367948at_nat
@ ^ [S2: produc5825016348098550007at_nat] : P2 )
= top_to4412277702576798813at_nat ) )
& ( ~ P2
=> ( ( collec7697611494764367948at_nat
@ ^ [S2: produc5825016348098550007at_nat] : P2 )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_const
thf(fact_297_finite__Collect__not,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat @ ( collec7697611494764367948at_nat @ P2 ) )
=> ( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
~ ( P2 @ X ) ) )
= ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ) ).
% finite_Collect_not
thf(fact_298_top__prod__def,axiom,
( top_to3998398741293409911od_o_o
= ( product_Pair_o_o @ top_top_o @ top_top_o ) ) ).
% top_prod_def
thf(fact_299_UNIV__eq__I,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( top_to4412277702576798813at_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_300_UNIV__eq__I,axiom,
! [A4: set_Product_prod_o_o] :
( ! [X3: product_prod_o_o] : ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( top_to7721136755696657239od_o_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_301_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_302_UNIV__witness,axiom,
? [X3: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X3 @ top_to4412277702576798813at_nat ) ).
% UNIV_witness
thf(fact_303_UNIV__witness,axiom,
? [X3: product_prod_o_o] : ( member7466972457876170832od_o_o @ X3 @ top_to7721136755696657239od_o_o ) ).
% UNIV_witness
thf(fact_304_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_305_Finite__Set_Ofinite__set,axiom,
( ( finite2560455063271695812at_nat @ top_to3676569147237732883at_nat )
= ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ).
% Finite_Set.finite_set
thf(fact_306_UNIV__def,axiom,
( top_to4412277702576798813at_nat
= ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : $true ) ) ).
% UNIV_def
thf(fact_307_finite__prod,axiom,
( ( finite8185985059073859960at_nat @ top_to461790979773641599at_nat )
= ( ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
& ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ) ).
% finite_prod
thf(fact_308_finite__prod,axiom,
( ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
= ( ( finite5600759454172676150la_a_b @ top_to8023889138469157253la_a_b )
& ( finite9047747110432174090at_nat @ top_to7629004291339433233at_nat ) ) ) ).
% finite_prod
thf(fact_309_finite__Prod__UNIV,axiom,
( ( finite5600759454172676150la_a_b @ top_to8023889138469157253la_a_b )
=> ( ( finite9047747110432174090at_nat @ top_to7629004291339433233at_nat )
=> ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_310_finite__Prod__UNIV,axiom,
( ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
=> ( ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
=> ( finite8185985059073859960at_nat @ top_to461790979773641599at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_311_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_312_empty__not__UNIV,axiom,
bot_bo7073875226086086771od_o_o != top_to7721136755696657239od_o_o ).
% empty_not_UNIV
thf(fact_313_empty__not__UNIV,axiom,
bot_bo1973934853101755969at_nat != top_to4412277702576798813at_nat ).
% empty_not_UNIV
thf(fact_314_ex__new__if__finite,axiom,
! [A4: set_Product_prod_o_o] :
( ~ ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o )
=> ( ( finite6120865539452801872od_o_o @ A4 )
=> ? [A2: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ A2 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_315_ex__new__if__finite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A4 )
=> ? [A2: nat] :
~ ( member_nat @ A2 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_316_ex__new__if__finite,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat )
=> ( ( finite6790245451575510286at_nat @ A4 )
=> ? [A2: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ A2 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_317_Id__onE,axiom,
! [C: produc364298880807886031at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ C @ ( id_on_89751120774077850at_nat @ A4 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( C
!= ( produc3112276593429206151at_nat @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_318_Id__onE,axiom,
! [C: produc4465838814804890855od_o_o,A4: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ C @ ( id_on_8897472890386745284od_o_o @ A4 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( C
!= ( produc3331439552374987607od_o_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_319_Id__onE,axiom,
! [C: product_prod_nat_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_320_Id__onE,axiom,
! [C: product_prod_o_o,A4: set_o] :
( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A4 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ( C
!= ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_321_Id__on__eqI,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( A = B )
=> ( ( member6956540099943067662at_nat @ A @ A4 )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ ( id_on_89751120774077850at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_322_Id__on__eqI,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( A = B )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( id_on_8897472890386745284od_o_o @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_323_Id__on__eqI,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_324_Id__on__eqI,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( A = B )
=> ( ( member_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( id_on_o @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_325_Id__on__iff,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( id_on_89751120774077850at_nat @ A4 ) )
= ( ( X4 = Y3 )
& ( member6956540099943067662at_nat @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_326_Id__on__iff,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( id_on_8897472890386745284od_o_o @ A4 ) )
= ( ( X4 = Y3 )
& ( member7466972457876170832od_o_o @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_327_Id__on__iff,axiom,
! [X4: nat,Y3: nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( id_on_nat @ A4 ) )
= ( ( X4 = Y3 )
& ( member_nat @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_328_Id__on__iff,axiom,
! [X4: $o,Y3: $o,A4: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( id_on_o @ A4 ) )
= ( ( X4 = Y3 )
& ( member_o @ X4 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_329_curryD,axiom,
! [F: produc5825016348098550007at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( produc4986730048175820513_nat_o @ F @ A @ B )
=> ( F @ ( produc760948067230524457at_nat @ A @ B ) ) ) ).
% curryD
thf(fact_330_curryD,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( product_curry_o_o_o @ F @ A @ B )
=> ( F @ ( product_Pair_o_o @ A @ B ) ) ) ).
% curryD
thf(fact_331_curryE,axiom,
! [F: produc5825016348098550007at_nat > $o,A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat] :
( ( produc4986730048175820513_nat_o @ F @ A @ B )
=> ( F @ ( produc760948067230524457at_nat @ A @ B ) ) ) ).
% curryE
thf(fact_332_curryE,axiom,
! [F: product_prod_o_o > $o,A: $o,B: $o] :
( ( product_curry_o_o_o @ F @ A @ B )
=> ( F @ ( product_Pair_o_o @ A @ B ) ) ) ).
% curryE
thf(fact_333_Id__on__def_H,axiom,
! [A4: produc5825016348098550007at_nat > $o] :
( ( id_on_89751120774077850at_nat @ ( collec7697611494764367948at_nat @ A4 ) )
= ( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ^ [X: produc5825016348098550007at_nat,Y4: produc5825016348098550007at_nat] :
( ( X = Y4 )
& ( A4 @ X ) ) ) ) ) ).
% Id_on_def'
thf(fact_334_Id__on__def_H,axiom,
! [A4: $o > $o] :
( ( id_on_o @ ( collect_o @ A4 ) )
= ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( X = Y4 )
& ( A4 @ X ) ) ) ) ) ).
% Id_on_def'
thf(fact_335_bijective__def,axiom,
( biject6085764940634361907at_nat
= ( ^ [R4: set_Pr2645174627780777389at_nat] :
( ! [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
( ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ R4 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Z3 ) @ R4 ) )
=> ( Y4 = Z3 ) )
& ! [X: relational_fmla_a_b,Y4: relational_fmla_a_b,Z3: set_Pr1261947904930325089at_nat] :
( ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Z3 ) @ R4 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ Y4 @ Z3 ) @ R4 ) )
=> ( X = Y4 ) ) ) ) ) ).
% bijective_def
thf(fact_336_bijective__def,axiom,
( bijective_o_o
= ( ^ [R4: set_Product_prod_o_o] :
( ! [X: $o,Y4: $o,Z3: $o] :
( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R4 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Z3 ) @ R4 ) )
=> ( Y4 = Z3 ) )
& ! [X: $o,Y4: $o,Z3: $o] :
( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Z3 ) @ R4 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ Z3 ) @ R4 ) )
=> ( X = Y4 ) ) ) ) ) ).
% bijective_def
thf(fact_337_finite__option__UNIV,axiom,
( ( finite8010495993044527070at_nat @ top_to8417037425451430189at_nat )
= ( finite6790245451575510286at_nat @ top_to4412277702576798813at_nat ) ) ).
% finite_option_UNIV
thf(fact_338_iso__tuple__UNIV__I,axiom,
! [X4: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X4 @ top_to4412277702576798813at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_339_iso__tuple__UNIV__I,axiom,
! [X4: product_prod_o_o] : ( member7466972457876170832od_o_o @ X4 @ top_to7721136755696657239od_o_o ) ).
% iso_tuple_UNIV_I
thf(fact_340_iso__tuple__UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_341_aboveS__def,axiom,
( order_2677216917948982476at_nat
= ( ^ [R2: set_Pr3205448351804440111at_nat,A3: produc5825016348098550007at_nat] :
( collec7697611494764367948at_nat
@ ^ [B3: produc5825016348098550007at_nat] :
( ( B3 != A3 )
& ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A3 @ B3 ) @ R2 ) ) ) ) ) ).
% aboveS_def
thf(fact_342_aboveS__def,axiom,
( order_aboveS_o
= ( ^ [R2: set_Product_prod_o_o,A3: $o] :
( collect_o
@ ^ [B3: $o] :
( ( B3 != A3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R2 ) ) ) ) ) ).
% aboveS_def
thf(fact_343_Sigma__UNIV__cancel,axiom,
! [A4: set_o,X5: set_o] :
( ( minus_7052513330764642158od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : X5 )
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : top_top_set_o ) )
= bot_bo7073875226086086771od_o_o ) ).
% Sigma_UNIV_cancel
thf(fact_344_Sigma__UNIV__cancel,axiom,
! [A4: set_Re381260168593705685la_a_b,X5: set_se7855581050983116737at_nat] :
( ( minus_6698950876951835142at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : X5 )
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : top_to7629004291339433233at_nat ) )
= bot_bo1973934853101755969at_nat ) ).
% Sigma_UNIV_cancel
thf(fact_345_above__def,axiom,
( order_635228650547596791at_nat
= ( ^ [R2: set_Pr3205448351804440111at_nat,A3: produc5825016348098550007at_nat] :
( collec7697611494764367948at_nat
@ ^ [B3: produc5825016348098550007at_nat] : ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A3 @ B3 ) @ R2 ) ) ) ) ).
% above_def
thf(fact_346_above__def,axiom,
( order_above_o
= ( ^ [R2: set_Product_prod_o_o,A3: $o] :
( collect_o
@ ^ [B3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R2 ) ) ) ) ).
% above_def
thf(fact_347_max__ext_Ocases,axiom,
! [A1: set_o,A22: set_o,R: set_Product_prod_o_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R ) )
=> ~ ( ( finite_finite_o @ A1 )
=> ( ( finite_finite_o @ A22 )
=> ( ( A22 != bot_bot_set_o )
=> ~ ! [X6: $o] :
( ( member_o @ X6 @ A1 )
=> ? [Xa2: $o] :
( ( member_o @ Xa2 @ A22 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_348_max__ext_Ocases,axiom,
! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
=> ~ ( ( finite_finite_nat @ A1 )
=> ( ( finite_finite_nat @ A22 )
=> ( ( A22 != bot_bot_set_nat )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ A1 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_349_max__ext_Ocases,axiom,
! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr7011910401827753159od_o_o] :
( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R ) )
=> ~ ( ( finite6120865539452801872od_o_o @ A1 )
=> ( ( finite6120865539452801872od_o_o @ A22 )
=> ( ( A22 != bot_bo7073875226086086771od_o_o )
=> ~ ! [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A1 )
=> ? [Xa2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa2 @ A22 )
& ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_350_max__ext_Ocases,axiom,
! [A1: set_Pr2645174627780777389at_nat,A22: set_Pr2645174627780777389at_nat,R: set_Pr3205448351804440111at_nat] :
( ( member8660133138740149732at_nat @ ( produc4461856759730394867at_nat @ A1 @ A22 ) @ ( max_ex5394131240172625916at_nat @ R ) )
=> ~ ( ( finite6790245451575510286at_nat @ A1 )
=> ( ( finite6790245451575510286at_nat @ A22 )
=> ( ( A22 != bot_bo1973934853101755969at_nat )
=> ~ ! [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ A1 )
=> ? [Xa2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa2 @ A22 )
& ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_351_max__ext_Osimps,axiom,
! [A1: set_o,A22: set_o,R: set_Product_prod_o_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R ) )
= ( ( finite_finite_o @ A1 )
& ( finite_finite_o @ A22 )
& ( A22 != bot_bot_set_o )
& ! [X: $o] :
( ( member_o @ X @ A1 )
=> ? [Y4: $o] :
( ( member_o @ Y4 @ A22 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_352_max__ext_Osimps,axiom,
! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
= ( ( finite_finite_nat @ A1 )
& ( finite_finite_nat @ A22 )
& ( A22 != bot_bot_set_nat )
& ! [X: nat] :
( ( member_nat @ X @ A1 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_353_max__ext_Osimps,axiom,
! [A1: set_Product_prod_o_o,A22: set_Product_prod_o_o,R: set_Pr7011910401827753159od_o_o] :
( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A1 @ A22 ) @ ( max_ex7572203241849361634od_o_o @ R ) )
= ( ( finite6120865539452801872od_o_o @ A1 )
& ( finite6120865539452801872od_o_o @ A22 )
& ( A22 != bot_bo7073875226086086771od_o_o )
& ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A1 )
=> ? [Y4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y4 @ A22 )
& ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X @ Y4 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_354_max__ext_Osimps,axiom,
! [A1: set_Pr2645174627780777389at_nat,A22: set_Pr2645174627780777389at_nat,R: set_Pr3205448351804440111at_nat] :
( ( member8660133138740149732at_nat @ ( produc4461856759730394867at_nat @ A1 @ A22 ) @ ( max_ex5394131240172625916at_nat @ R ) )
= ( ( finite6790245451575510286at_nat @ A1 )
& ( finite6790245451575510286at_nat @ A22 )
& ( A22 != bot_bo1973934853101755969at_nat )
& ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A1 )
=> ? [Y4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y4 @ A22 )
& ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X @ Y4 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_355_Diff__iff,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) )
= ( ( member6956540099943067662at_nat @ C @ A4 )
& ~ ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_356_Diff__iff,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ C @ A4 )
& ~ ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_357_Diff__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_358_DiffI,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ A4 )
=> ( ~ ( member6956540099943067662at_nat @ C @ B5 )
=> ( member6956540099943067662at_nat @ C @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_359_DiffI,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ C @ B5 )
=> ( member7466972457876170832od_o_o @ C @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_360_DiffI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_361_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_362_Diff__cancel,axiom,
! [A4: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ A4 @ A4 )
= bot_bo7073875226086086771od_o_o ) ).
% Diff_cancel
thf(fact_363_Diff__cancel,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A4 @ A4 )
= bot_bo1973934853101755969at_nat ) ).
% Diff_cancel
thf(fact_364_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_365_empty__Diff,axiom,
! [A4: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ bot_bo7073875226086086771od_o_o @ A4 )
= bot_bo7073875226086086771od_o_o ) ).
% empty_Diff
thf(fact_366_empty__Diff,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ bot_bo1973934853101755969at_nat @ A4 )
= bot_bo1973934853101755969at_nat ) ).
% empty_Diff
thf(fact_367_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_368_Diff__empty,axiom,
! [A4: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ A4 @ bot_bo7073875226086086771od_o_o )
= A4 ) ).
% Diff_empty
thf(fact_369_Diff__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A4 @ bot_bo1973934853101755969at_nat )
= A4 ) ).
% Diff_empty
thf(fact_370_finite__Diff2,axiom,
! [B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ B5 )
=> ( ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) )
= ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_371_finite__Diff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_372_Diff__UNIV,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_373_Diff__UNIV,axiom,
! [A4: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ A4 @ top_to7721136755696657239od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Diff_UNIV
thf(fact_374_Diff__UNIV,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A4 @ top_to4412277702576798813at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Diff_UNIV
thf(fact_375_DiffD2,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) )
=> ~ ( member6956540099943067662at_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_376_DiffD2,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) )
=> ~ ( member7466972457876170832od_o_o @ C @ B5 ) ) ).
% DiffD2
thf(fact_377_DiffD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_378_DiffD1,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) )
=> ( member6956540099943067662at_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_379_DiffD1,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) )
=> ( member7466972457876170832od_o_o @ C @ A4 ) ) ).
% DiffD1
thf(fact_380_DiffD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_381_DiffE,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) )
=> ~ ( ( member6956540099943067662at_nat @ C @ A4 )
=> ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_382_DiffE,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) )
=> ~ ( ( member7466972457876170832od_o_o @ C @ A4 )
=> ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% DiffE
thf(fact_383_DiffE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_384_set__diff__eq,axiom,
( minus_7052513330764642158od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A7 )
& ~ ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_385_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A7 )
& ~ ( member_nat @ X @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_386_set__diff__eq,axiom,
( minus_6698950876951835142at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A7 )
& ~ ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_387_top__empty__eq,axiom,
( top_to7031512898858585704_nat_o
= ( ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ top_to4412277702576798813at_nat ) ) ) ).
% top_empty_eq
thf(fact_388_top__empty__eq,axiom,
( top_to1116032131123568710_o_o_o
= ( ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ top_to7721136755696657239od_o_o ) ) ) ).
% top_empty_eq
thf(fact_389_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_390_top__set__def,axiom,
( top_to4412277702576798813at_nat
= ( collec7697611494764367948at_nat @ top_to7031512898858585704_nat_o ) ) ).
% top_set_def
thf(fact_391_Diff__infinite__finite,axiom,
! [T: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ T )
=> ( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_392_finite__if__eq__beyond__finite,axiom,
! [S: set_Pr2645174627780777389at_nat,S3: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ S )
=> ( finite2560455063271695812at_nat
@ ( collec5061078246732527874at_nat
@ ^ [S2: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ S2 @ S )
= ( minus_6698950876951835142at_nat @ S3 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_393_top__empty__eq2,axiom,
( top_to2171089622161864010_nat_o
= ( ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ top_to4412277702576798813at_nat ) ) ) ).
% top_empty_eq2
thf(fact_394_top__empty__eq2,axiom,
( top_top_o_o_o
= ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ top_to7721136755696657239od_o_o ) ) ) ).
% top_empty_eq2
thf(fact_395_max__ext_Omax__extI,axiom,
! [X5: set_o,Y6: set_o,R: set_Product_prod_o_o] :
( ( finite_finite_o @ X5 )
=> ( ( finite_finite_o @ Y6 )
=> ( ( Y6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ? [Xa: $o] :
( ( member_o @ Xa @ Y6 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Xa ) @ R ) ) )
=> ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X5 @ Y6 ) @ ( max_ext_o @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_396_max__ext_Omax__extI,axiom,
! [X5: set_nat,Y6: set_nat,R: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( Y6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ Y6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa ) @ R ) ) )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y6 ) @ ( max_ext_nat @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_397_max__ext_Omax__extI,axiom,
! [X5: set_Product_prod_o_o,Y6: set_Product_prod_o_o,R: set_Pr7011910401827753159od_o_o] :
( ( finite6120865539452801872od_o_o @ X5 )
=> ( ( finite6120865539452801872od_o_o @ Y6 )
=> ( ( Y6 != bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ X5 )
=> ? [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ Y6 )
& ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X3 @ Xa ) @ R ) ) )
=> ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X5 @ Y6 ) @ ( max_ex7572203241849361634od_o_o @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_398_max__ext_Omax__extI,axiom,
! [X5: set_Pr2645174627780777389at_nat,Y6: set_Pr2645174627780777389at_nat,R: set_Pr3205448351804440111at_nat] :
( ( finite6790245451575510286at_nat @ X5 )
=> ( ( finite6790245451575510286at_nat @ Y6 )
=> ( ( Y6 != bot_bo1973934853101755969at_nat )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ X5 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ Y6 )
& ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X3 @ Xa ) @ R ) ) )
=> ( member8660133138740149732at_nat @ ( produc4461856759730394867at_nat @ X5 @ Y6 ) @ ( max_ex5394131240172625916at_nat @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_399_max__extp__max__ext__eq,axiom,
! [R: set_Product_prod_o_o] :
( ( max_extp_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R ) )
= ( ^ [X: set_o,Y4: set_o] : ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X @ Y4 ) @ ( max_ext_o @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_400_max__ext__def,axiom,
( max_ext_o
= ( ^ [R4: set_Product_prod_o_o] :
( collec1488362387706010770_set_o
@ ( produc8987465890351537004et_o_o
@ ( max_extp_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R4 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_401_symcl__def,axiom,
( symcl_o
= ( ^ [R2: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R2 )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X ) @ R2 ) ) ) ) ) ) ).
% symcl_def
thf(fact_402_max__extp__eq,axiom,
( max_extp_o
= ( ^ [R2: $o > $o > $o,X: set_o,Y4: set_o] : ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X @ Y4 ) @ ( max_ext_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ R2 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_403_rtrancl__last__touch,axiom,
! [Q2: produc5825016348098550007at_nat,Q3: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Q3 ) @ ( transi7565848034366790580at_nat @ R ) )
=> ( ( member6956540099943067662at_nat @ Q2 @ S )
=> ~ ! [Qt: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Qt @ S )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Qt ) @ ( transi7565848034366790580at_nat @ R ) )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Qt @ Q3 )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ R
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_404_rtrancl__last__touch,axiom,
! [Q2: product_prod_o_o,Q3: product_prod_o_o,R: set_Pr7011910401827753159od_o_o,S: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Q3 ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ( ( member7466972457876170832od_o_o @ Q2 @ S )
=> ~ ! [Qt: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Qt @ S )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Qt ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Qt @ Q3 )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ R
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_405_rtrancl__last__touch,axiom,
! [Q2: nat,Q3: nat,R: set_Pr1261947904930325089at_nat,S: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Q3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member_nat @ Q2 @ S )
=> ~ ! [Qt: nat] :
( ( member_nat @ Qt @ S )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Qt ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Qt @ Q3 )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ R
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_406_rtrancl__last__touch,axiom,
! [Q2: $o,Q3: $o,R: set_Product_prod_o_o,S: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Q3 ) @ ( transitive_rtrancl_o @ R ) )
=> ( ( member_o @ Q2 @ S )
=> ~ ! [Qt: $o] :
( ( member_o @ Qt @ S )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Qt ) @ ( transitive_rtrancl_o @ R ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Qt @ Q3 )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ R
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_407_rtrancl__last__visit_H,axiom,
! [Q2: produc5825016348098550007at_nat,Q3: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Q3 ) @ ( transi7565848034366790580at_nat @ R ) )
=> ( ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Q3 )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ R
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) )
=> ~ ! [Qt: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Qt @ S )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Qt ) @ ( transi7565848034366790580at_nat @ R ) )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Qt @ Q3 )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ R
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_408_rtrancl__last__visit_H,axiom,
! [Q2: product_prod_o_o,Q3: product_prod_o_o,R: set_Pr7011910401827753159od_o_o,S: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Q3 ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ( ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Q3 )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ R
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : S ) ) ) )
=> ~ ! [Qt: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Qt @ S )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Qt ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Qt @ Q3 )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ R
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_409_rtrancl__last__visit_H,axiom,
! [Q2: nat,Q3: nat,R: set_Pr1261947904930325089at_nat,S: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Q3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Q3 )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ R
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : S ) ) ) )
=> ~ ! [Qt: nat] :
( ( member_nat @ Qt @ S )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Qt ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Qt @ Q3 )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ R
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_410_rtrancl__last__visit_H,axiom,
! [Q2: $o,Q3: $o,R: set_Product_prod_o_o,S: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Q3 ) @ ( transitive_rtrancl_o @ R ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Q3 )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ R
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : S ) ) ) )
=> ~ ! [Qt: $o] :
( ( member_o @ Qt @ S )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Qt ) @ ( transitive_rtrancl_o @ R ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Qt @ Q3 )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ R
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_411_lex__prod__def,axiom,
( lex_pr2837670122506411537at_nat
= ( ^ [Ra: set_Pr389361089297601407la_a_b,Rb: set_Pr4329608150637261639at_nat] :
( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ( produc1661938560264169794_nat_o
@ ^ [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat] :
( produc5861013030283631554_nat_o
@ ^ [A8: relational_fmla_a_b,B8: set_Pr1261947904930325089at_nat] :
( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A3 @ A8 ) @ Ra )
| ( ( A3 = A8 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_412_lex__prod__def,axiom,
( lex_prod_o_o
= ( ^ [Ra: set_Product_prod_o_o,Rb: set_Product_prod_o_o] :
( collec7923239488154759506od_o_o
@ ( produc7860636428938663212_o_o_o
@ ( produc5911696763256879298_o_o_o
@ ^ [A3: $o,B3: $o] :
( produc6197397395684419436_o_o_o
@ ^ [A8: $o,B8: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A8 ) @ Ra )
| ( ( A3 = A8 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_413_in__lex__prod,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,A5: relational_fmla_a_b,B4: set_Pr1261947904930325089at_nat,R3: set_Pr389361089297601407la_a_b,S4: set_Pr4329608150637261639at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ ( produc760948067230524457at_nat @ A5 @ B4 ) ) @ ( lex_pr2837670122506411537at_nat @ R3 @ S4 ) )
= ( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B4 ) @ S4 ) ) ) ) ).
% in_lex_prod
thf(fact_414_in__lex__prod,axiom,
! [A: $o,B: $o,A5: $o,B4: $o,R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Pair_o_o @ A5 @ B4 ) ) @ ( lex_prod_o_o @ R3 @ S4 ) )
= ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A5 ) @ R3 )
| ( ( A = A5 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ B4 ) @ S4 ) ) ) ) ).
% in_lex_prod
thf(fact_415_converse__rtranclE_H,axiom,
! [U: $o,V: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ R ) )
=> ( ( U = (~ V) )
=> ~ ! [Vh: $o] :
( ( U = (~ Vh) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ Vh ) @ R )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Vh @ V ) @ ( transitive_rtrancl_o @ R ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_416_minus__set__def,axiom,
( minus_7052513330764642158od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ( minus_7418568063025027823_o_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A7 )
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_417_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A7 )
@ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_418_minus__set__def,axiom,
( minus_6698950876951835142at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ( minus_8035352901537135103_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A7 )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_419_in__symclI_I2_J,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( symcl_o @ R3 ) ) ) ).
% in_symclI(2)
thf(fact_420_in__symclI_I1_J,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( symcl_o @ R3 ) ) ) ).
% in_symclI(1)
thf(fact_421_converse__rtrancl__into__rtrancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_422_converse__rtrancl__induct,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,P2: $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( P2 @ B )
=> ( ! [Y: $o,Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_423_converse__rtranclE,axiom,
! [X4: $o,Z: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( X4 = (~ Z) )
=> ~ ! [Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_424_rtrancl__induct,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,P2: $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( P2 @ A )
=> ( ! [Y: $o,Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ Y ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ R3 )
=> ( ( P2 @ Y )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_425_rtrancl__trans,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,Z: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_426_rtranclE,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( A = (~ B) )
=> ~ ! [Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ Y ) @ ( transitive_rtrancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ B ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_427_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_428_rtrancl_Ortrancl__refl,axiom,
! [A: $o,R3: set_Product_prod_o_o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( transitive_rtrancl_o @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_429_rtrancl__induct2,axiom,
! [Ax: relational_fmla_a_b,Ay: set_Pr1261947904930325089at_nat,Bx: relational_fmla_a_b,By: set_Pr1261947904930325089at_nat,R3: set_Pr3205448351804440111at_nat,P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ Bx @ By ) ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,Aa: relational_fmla_a_b,Ba: set_Pr1261947904930325089at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ A2 @ B2 ) ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ A2 @ B2 ) @ ( produc760948067230524457at_nat @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A2 @ B2 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_430_rtrancl__induct2,axiom,
! [Ax: $o,Ay: $o,Bx: $o,By: $o,R3: set_Pr7011910401827753159od_o_o,P2: $o > $o > $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ Bx @ By ) ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A2: $o,B2: $o,Aa: $o,Ba: $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ A2 @ B2 ) ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ ( product_Pair_o_o @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A2 @ B2 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_431_converse__rtranclE2,axiom,
! [Xa3: relational_fmla_a_b,Xb: set_Pr1261947904930325089at_nat,Za: relational_fmla_a_b,Zb: set_Pr1261947904930325089at_nat,R3: set_Pr3205448351804440111at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Xa3 @ Xb ) @ ( produc760948067230524457at_nat @ Za @ Zb ) ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( ( produc760948067230524457at_nat @ Xa3 @ Xb )
!= ( produc760948067230524457at_nat @ Za @ Zb ) )
=> ~ ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Xa3 @ Xb ) @ ( produc760948067230524457at_nat @ A2 @ B2 ) ) @ R3 )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ A2 @ B2 ) @ ( produc760948067230524457at_nat @ Za @ Zb ) ) @ ( transi7565848034366790580at_nat @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_432_converse__rtranclE2,axiom,
! [Xa3: $o,Xb: $o,Za: $o,Zb: $o,R3: set_Pr7011910401827753159od_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Xa3 @ Xb ) @ ( product_Pair_o_o @ Za @ Zb ) ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( ( product_Pair_o_o @ Xa3 @ Xb )
!= ( product_Pair_o_o @ Za @ Zb ) )
=> ~ ! [A2: $o,B2: $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Xa3 @ Xb ) @ ( product_Pair_o_o @ A2 @ B2 ) ) @ R3 )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ ( product_Pair_o_o @ Za @ Zb ) ) @ ( transi4448435615503354666od_o_o @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_433_converse__rtrancl__induct2,axiom,
! [Ax: relational_fmla_a_b,Ay: set_Pr1261947904930325089at_nat,Bx: relational_fmla_a_b,By: set_Pr1261947904930325089at_nat,R3: set_Pr3205448351804440111at_nat,P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ Bx @ By ) ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,Aa: relational_fmla_a_b,Ba: set_Pr1261947904930325089at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ A2 @ B2 ) @ ( produc760948067230524457at_nat @ Aa @ Ba ) ) @ R3 )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Aa @ Ba ) @ ( produc760948067230524457at_nat @ Bx @ By ) ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A2 @ B2 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_434_converse__rtrancl__induct2,axiom,
! [Ax: $o,Ay: $o,Bx: $o,By: $o,R3: set_Pr7011910401827753159od_o_o,P2: $o > $o > $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ Bx @ By ) ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A2: $o,B2: $o,Aa: $o,Ba: $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ ( product_Pair_o_o @ Aa @ Ba ) ) @ R3 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Aa @ Ba ) @ ( product_Pair_o_o @ Bx @ By ) ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A2 @ B2 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_435_rtrancl_Ocases,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( A22 = (~ A1) )
=> ~ ! [B2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B2 ) @ ( transitive_rtrancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_436_rtrancl_Osimps,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( transitive_rtrancl_o @ R3 ) )
= ( ? [A3: $o] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: $o,B3: $o,C3: $o] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ ( transitive_rtrancl_o @ R3 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ C3 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_437_same__fst__def,axiom,
( same_f9172395308798353812at_nat
= ( ^ [P4: relational_fmla_a_b > $o,R4: relational_fmla_a_b > set_Pr4329608150637261639at_nat] :
( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ( produc1661938560264169794_nat_o
@ ^ [X7: relational_fmla_a_b,Y7: set_Pr1261947904930325089at_nat] :
( produc5861013030283631554_nat_o
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] :
( ( X7 = X )
& ( P4 @ X )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_438_same__fst__def,axiom,
( same_fst_o_o
= ( ^ [P4: $o > $o,R4: $o > set_Product_prod_o_o] :
( collec7923239488154759506od_o_o
@ ( produc7860636428938663212_o_o_o
@ ( produc5911696763256879298_o_o_o
@ ^ [X7: $o,Y7: $o] :
( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( X7 = X )
& ( P4 @ X )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y7 @ Y4 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_439_rtrancl__restrictI,axiom,
! [U: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ E
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : R ) ) ) )
=> ( ~ ( member6956540099943067662at_nat @ U @ R )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ ( transi7565848034366790580at_nat @ ( rel_re178346425959758229at_nat @ E @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_440_rtrancl__restrictI,axiom,
! [U: product_prod_o_o,V: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ E
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : R ) ) ) )
=> ( ~ ( member7466972457876170832od_o_o @ U @ R )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ ( transi4448435615503354666od_o_o @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_441_rtrancl__restrictI,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ E
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : R ) ) ) )
=> ( ~ ( member_nat @ U @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ ( rel_restrict_nat @ E @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_442_rtrancl__restrictI,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ E
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : R ) ) ) )
=> ( ~ ( member_o @ U @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ ( rel_restrict_o @ E @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_443_rtrancl__last__visit,axiom,
! [Q2: produc5825016348098550007at_nat,Q3: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Q3 ) @ ( transi7565848034366790580at_nat @ R ) )
=> ( ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Q3 )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ R
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) )
=> ~ ! [Qt: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Qt @ S )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ Qt ) @ ( transi5629900152392516758at_nat @ R ) )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Qt @ Q3 )
@ ( transi7565848034366790580at_nat
@ ( minus_765089489760238870at_nat @ R
@ ( produc984725824077784504at_nat @ top_to4412277702576798813at_nat
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_444_rtrancl__last__visit,axiom,
! [Q2: product_prod_o_o,Q3: product_prod_o_o,R: set_Pr7011910401827753159od_o_o,S: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Q3 ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ( ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Q3 )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ R
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : S ) ) ) )
=> ~ ! [Qt: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Qt @ S )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ Qt ) @ ( transi1494748803203666376od_o_o @ R ) )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Qt @ Q3 )
@ ( transi4448435615503354666od_o_o
@ ( minus_1612342758575478574od_o_o @ R
@ ( produc6514336722952383752od_o_o @ top_to7721136755696657239od_o_o
@ ^ [Uu: product_prod_o_o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_445_rtrancl__last__visit,axiom,
! [Q2: nat,Q3: nat,R: set_Pr1261947904930325089at_nat,S: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Q3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Q3 )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ R
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : S ) ) ) )
=> ~ ! [Qt: nat] :
( ( member_nat @ Qt @ S )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ Qt ) @ ( transi6264000038957366511cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Qt @ Q3 )
@ ( transi2905341329935302413cl_nat
@ ( minus_1356011639430497352at_nat @ R
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_446_rtrancl__last__visit,axiom,
! [Q2: $o,Q3: $o,R: set_Product_prod_o_o,S: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Q3 ) @ ( transitive_rtrancl_o @ R ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Q3 )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ R
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : S ) ) ) )
=> ~ ! [Qt: $o] :
( ( member_o @ Qt @ S )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ Qt ) @ ( transitive_trancl_o @ R ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Qt @ Q3 )
@ ( transitive_rtrancl_o
@ ( minus_7052513330764642158od_o_o @ R
@ ( product_Sigma_o_o @ top_top_set_o
@ ^ [Uu: $o] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_447_max__extp_Omax__extI,axiom,
! [X5: set_Product_prod_o_o,Y6: set_Product_prod_o_o,R: product_prod_o_o > product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ X5 )
=> ( ( finite6120865539452801872od_o_o @ Y6 )
=> ( ( Y6
!= ( collec3167064739498627218od_o_o @ bot_bo6429049293326997034_o_o_o ) )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ X5 )
=> ? [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ Y6 )
& ( R @ X3 @ Xa ) ) )
=> ( max_ex859744148169958202od_o_o @ R @ X5 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_448_max__extp_Omax__extI,axiom,
! [X5: set_nat,Y6: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( Y6
!= ( collect_nat @ bot_bot_nat_o ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ Y6 )
& ( R @ X3 @ Xa ) ) )
=> ( max_extp_nat @ R @ X5 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_449_max__extp_Omax__extI,axiom,
! [X5: set_Pr2645174627780777389at_nat,Y6: set_Pr2645174627780777389at_nat,R: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat @ X5 )
=> ( ( finite6790245451575510286at_nat @ Y6 )
=> ( ( Y6
!= ( collec7697611494764367948at_nat @ bot_bo3884824731534426500_nat_o ) )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ X5 )
=> ? [Xa: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa @ Y6 )
& ( R @ X3 @ Xa ) ) )
=> ( max_ex4591110627596355492at_nat @ R @ X5 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_450_max__extp_Osimps,axiom,
( max_ex859744148169958202od_o_o
= ( ^ [R4: product_prod_o_o > product_prod_o_o > $o,A12: set_Product_prod_o_o,A23: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A12 )
& ( finite6120865539452801872od_o_o @ A23 )
& ( A23
!= ( collec3167064739498627218od_o_o @ bot_bo6429049293326997034_o_o_o ) )
& ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A12 )
=> ? [Y4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y4 @ A23 )
& ( R4 @ X @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_451_max__extp_Osimps,axiom,
( max_extp_nat
= ( ^ [R4: nat > nat > $o,A12: set_nat,A23: set_nat] :
( ( finite_finite_nat @ A12 )
& ( finite_finite_nat @ A23 )
& ( A23
!= ( collect_nat @ bot_bot_nat_o ) )
& ! [X: nat] :
( ( member_nat @ X @ A12 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ A23 )
& ( R4 @ X @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_452_max__extp_Osimps,axiom,
( max_ex4591110627596355492at_nat
= ( ^ [R4: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o,A12: set_Pr2645174627780777389at_nat,A23: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A12 )
& ( finite6790245451575510286at_nat @ A23 )
& ( A23
!= ( collec7697611494764367948at_nat @ bot_bo3884824731534426500_nat_o ) )
& ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A12 )
=> ? [Y4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y4 @ A23 )
& ( R4 @ X @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_453_max__extp_Ocases,axiom,
! [R: product_prod_o_o > product_prod_o_o > $o,A1: set_Product_prod_o_o,A22: set_Product_prod_o_o] :
( ( max_ex859744148169958202od_o_o @ R @ A1 @ A22 )
=> ~ ( ( finite6120865539452801872od_o_o @ A1 )
=> ( ( finite6120865539452801872od_o_o @ A22 )
=> ( ( A22
!= ( collec3167064739498627218od_o_o @ bot_bo6429049293326997034_o_o_o ) )
=> ~ ! [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A1 )
=> ? [Xa2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa2 @ A22 )
& ( R @ X6 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_454_max__extp_Ocases,axiom,
! [R: nat > nat > $o,A1: set_nat,A22: set_nat] :
( ( max_extp_nat @ R @ A1 @ A22 )
=> ~ ( ( finite_finite_nat @ A1 )
=> ( ( finite_finite_nat @ A22 )
=> ( ( A22
!= ( collect_nat @ bot_bot_nat_o ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ A1 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ A22 )
& ( R @ X6 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_455_max__extp_Ocases,axiom,
! [R: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o,A1: set_Pr2645174627780777389at_nat,A22: set_Pr2645174627780777389at_nat] :
( ( max_ex4591110627596355492at_nat @ R @ A1 @ A22 )
=> ~ ( ( finite6790245451575510286at_nat @ A1 )
=> ( ( finite6790245451575510286at_nat @ A22 )
=> ( ( A22
!= ( collec7697611494764367948at_nat @ bot_bo3884824731534426500_nat_o ) )
=> ~ ! [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ A1 )
=> ? [Xa2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Xa2 @ A22 )
& ( R @ X6 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_456_preorder__on__empty,axiom,
order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% preorder_on_empty
thf(fact_457_preorder__on__empty,axiom,
order_4861654808422542329on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% preorder_on_empty
thf(fact_458_preorder__on__empty,axiom,
order_3385673970029650238od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo2881837355931550515od_o_o ).
% preorder_on_empty
thf(fact_459_preorder__on__empty,axiom,
order_6739036243672267424at_nat @ bot_bo1973934853101755969at_nat @ bot_bo2147731989722242715at_nat ).
% preorder_on_empty
thf(fact_460_bex__empty,axiom,
! [P2: nat > $o] :
~ ? [X6: nat] :
( ( member_nat @ X6 @ bot_bot_set_nat )
& ( P2 @ X6 ) ) ).
% bex_empty
thf(fact_461_bex__empty,axiom,
! [P2: product_prod_o_o > $o] :
~ ? [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ bot_bo7073875226086086771od_o_o )
& ( P2 @ X6 ) ) ).
% bex_empty
thf(fact_462_bex__empty,axiom,
! [P2: produc5825016348098550007at_nat > $o] :
~ ? [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ bot_bo1973934853101755969at_nat )
& ( P2 @ X6 ) ) ).
% bex_empty
thf(fact_463_trancl__empty,axiom,
( ( transitive_trancl_o @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% trancl_empty
thf(fact_464_rel__restrict__empty,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( rel_restrict_nat @ R @ bot_bot_set_nat )
= R ) ).
% rel_restrict_empty
thf(fact_465_rel__restrict__empty,axiom,
! [R: set_Pr7011910401827753159od_o_o] :
( ( rel_re6977384794829015305od_o_o @ R @ bot_bo7073875226086086771od_o_o )
= R ) ).
% rel_restrict_empty
thf(fact_466_rel__restrict__empty,axiom,
! [R: set_Pr3205448351804440111at_nat] :
( ( rel_re178346425959758229at_nat @ R @ bot_bo1973934853101755969at_nat )
= R ) ).
% rel_restrict_empty
thf(fact_467_finite__Collect__bex,axiom,
! [A4: set_Pr2645174627780777389at_nat,Q: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
? [Y4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y4 @ A4 )
& ( Q @ X @ Y4 ) ) ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
=> ( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ^ [Y4: produc5825016348098550007at_nat] : ( Q @ Y4 @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_468_same__fstI,axiom,
! [P2: relational_fmla_a_b > $o,X4: relational_fmla_a_b,Y8: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R: relational_fmla_a_b > set_Pr4329608150637261639at_nat] :
( ( P2 @ X4 )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y8 @ Y3 ) @ ( R @ X4 ) )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ X4 @ Y8 ) @ ( produc760948067230524457at_nat @ X4 @ Y3 ) ) @ ( same_f9172395308798353812at_nat @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_469_same__fstI,axiom,
! [P2: $o > $o,X4: $o,Y8: $o,Y3: $o,R: $o > set_Product_prod_o_o] :
( ( P2 @ X4 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y8 @ Y3 ) @ ( R @ X4 ) )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ X4 @ Y8 ) @ ( product_Pair_o_o @ X4 @ Y3 ) ) @ ( same_fst_o_o @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_470_rel__restrict__trancl__notR_I2_J,axiom,
! [V: produc5825016348098550007at_nat,W: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ V @ W ) @ ( transi5629900152392516758at_nat @ ( rel_re178346425959758229at_nat @ E @ R ) ) )
=> ~ ( member6956540099943067662at_nat @ W @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_471_rel__restrict__trancl__notR_I2_J,axiom,
! [V: product_prod_o_o,W: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ V @ W ) @ ( transi1494748803203666376od_o_o @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) )
=> ~ ( member7466972457876170832od_o_o @ W @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_472_rel__restrict__trancl__notR_I2_J,axiom,
! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R ) ) )
=> ~ ( member_nat @ W @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_473_rel__restrict__trancl__notR_I2_J,axiom,
! [V: $o,W: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R ) ) )
=> ~ ( member_o @ W @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_474_rel__restrict__trancl__notR_I1_J,axiom,
! [V: produc5825016348098550007at_nat,W: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ V @ W ) @ ( transi5629900152392516758at_nat @ ( rel_re178346425959758229at_nat @ E @ R ) ) )
=> ~ ( member6956540099943067662at_nat @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_475_rel__restrict__trancl__notR_I1_J,axiom,
! [V: product_prod_o_o,W: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ V @ W ) @ ( transi1494748803203666376od_o_o @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) )
=> ~ ( member7466972457876170832od_o_o @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_476_rel__restrict__trancl__notR_I1_J,axiom,
! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R ) ) )
=> ~ ( member_nat @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_477_rel__restrict__trancl__notR_I1_J,axiom,
! [V: $o,W: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R ) ) )
=> ~ ( member_o @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_478_rel__restrict__trancl__mem,axiom,
! [A: $o,B: $o,A4: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ ( rel_restrict_o @ A4 @ R ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( rel_restrict_o @ ( transitive_trancl_o @ A4 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_479_Bex__def,axiom,
( bex_Pr1653138805792344471at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,P4: produc5825016348098550007at_nat > $o] :
? [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A7 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_480_Bex__def,axiom,
( bex_Product_prod_o_o
= ( ^ [A7: set_Product_prod_o_o,P4: product_prod_o_o > $o] :
? [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A7 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_481_Bex__def,axiom,
( bex_nat
= ( ^ [A7: set_nat,P4: nat > $o] :
? [X: nat] :
( ( member_nat @ X @ A7 )
& ( P4 @ X ) ) ) ) ).
% Bex_def
thf(fact_482_converse__trancl__induct,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,P2: $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ! [Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ B ) @ R3 )
=> ( P2 @ Y ) )
=> ( ! [Y: $o,Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_483_trancl__trans__induct,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,P2: $o > $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ R3 ) )
=> ( ! [X3: $o,Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ R3 )
=> ( P2 @ X3 @ Y ) )
=> ( ! [X3: $o,Y: $o,Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( P2 @ X3 @ Y )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( P2 @ Y @ Z2 )
=> ( P2 @ X3 @ Z2 ) ) ) ) )
=> ( P2 @ X4 @ Y3 ) ) ) ) ).
% trancl_trans_induct
thf(fact_484_trancl__into__trancl2,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( transitive_trancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_485_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_486_irrefl__trancl__rD,axiom,
! [R3: set_Product_prod_o_o,X4: $o,Y3: $o] :
( ! [X3: $o] :
~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R3 )
=> ( X4 != Y3 ) ) ) ).
% irrefl_trancl_rD
thf(fact_487_converse__tranclE,axiom,
! [X4: $o,Z: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transitive_trancl_o @ R3 ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ R3 )
=> ~ ! [Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z ) @ ( transitive_trancl_o @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_488_r__r__into__trancl,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_489_trancl__induct,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,P2: $o > $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ! [Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ Y ) @ R3 )
=> ( P2 @ Y ) )
=> ( ! [Y: $o,Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ Y ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ R3 )
=> ( ( P2 @ Y )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_490_trancl__trans,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,Z: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z ) @ ( transitive_trancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_491_tranclE,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ~ ! [C4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C4 ) @ ( transitive_trancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ C4 @ B ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_492_trancl_Or__into__trancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_493_trancl_Osimps,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( transitive_trancl_o @ R3 ) )
= ( ? [A3: $o,B3: $o] :
( ( A1 = A3 )
& ( A22 = B3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R3 ) )
| ? [A3: $o,B3: $o,C3: $o] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ ( transitive_trancl_o @ R3 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ C3 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_494_trancl_Ocases,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( transitive_trancl_o @ R3 ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ R3 )
=> ~ ! [B2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B2 ) @ ( transitive_trancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_495_rel__restrict__notR_I2_J,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A4: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( rel_re178346425959758229at_nat @ A4 @ R ) )
=> ~ ( member6956540099943067662at_nat @ Y3 @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_496_rel__restrict__notR_I2_J,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,A4: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( rel_re6977384794829015305od_o_o @ A4 @ R ) )
=> ~ ( member7466972457876170832od_o_o @ Y3 @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_497_rel__restrict__notR_I2_J,axiom,
! [X4: nat,Y3: nat,A4: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( rel_restrict_nat @ A4 @ R ) )
=> ~ ( member_nat @ Y3 @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_498_rel__restrict__notR_I2_J,axiom,
! [X4: $o,Y3: $o,A4: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( rel_restrict_o @ A4 @ R ) )
=> ~ ( member_o @ Y3 @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_499_rel__restrict__notR_I1_J,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A4: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( rel_re178346425959758229at_nat @ A4 @ R ) )
=> ~ ( member6956540099943067662at_nat @ X4 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_500_rel__restrict__notR_I1_J,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,A4: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( rel_re6977384794829015305od_o_o @ A4 @ R ) )
=> ~ ( member7466972457876170832od_o_o @ X4 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_501_rel__restrict__notR_I1_J,axiom,
! [X4: nat,Y3: nat,A4: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( rel_restrict_nat @ A4 @ R ) )
=> ~ ( member_nat @ X4 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_502_rel__restrict__notR_I1_J,axiom,
! [X4: $o,Y3: $o,A4: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( rel_restrict_o @ A4 @ R ) )
=> ~ ( member_o @ X4 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_503_rel__restrictI,axiom,
! [X4: produc5825016348098550007at_nat,R: set_Pr2645174627780777389at_nat,Y3: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ R )
=> ( ~ ( member6956540099943067662at_nat @ Y3 @ R )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ E )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( rel_re178346425959758229at_nat @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_504_rel__restrictI,axiom,
! [X4: product_prod_o_o,R: set_Product_prod_o_o,Y3: product_prod_o_o,E: set_Pr7011910401827753159od_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ R )
=> ( ~ ( member7466972457876170832od_o_o @ Y3 @ R )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ E )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_505_rel__restrictI,axiom,
! [X4: nat,R: set_nat,Y3: nat,E: set_Pr1261947904930325089at_nat] :
( ~ ( member_nat @ X4 @ R )
=> ( ~ ( member_nat @ Y3 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ E )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( rel_restrict_nat @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_506_rel__restrictI,axiom,
! [X4: $o,R: set_o,Y3: $o,E: set_Product_prod_o_o] :
( ~ ( member_o @ X4 @ R )
=> ( ~ ( member_o @ Y3 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ E )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( rel_restrict_o @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_507_rel__restrict__lift,axiom,
! [X4: $o,Y3: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( rel_restrict_o @ E @ R ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ E ) ) ).
% rel_restrict_lift
thf(fact_508_tranclD,axiom,
! [X4: $o,Y3: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ R ) )
=> ? [Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z2 ) @ R )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ Y3 ) @ ( transitive_rtrancl_o @ R ) ) ) ) ).
% tranclD
thf(fact_509_rtranclD,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R ) )
=> ( ( A = B )
| ( ( A != B )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R ) ) ) ) ) ).
% rtranclD
thf(fact_510_tranclD2,axiom,
! [X4: $o,Y3: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ R ) )
=> ? [Z2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z2 ) @ ( transitive_rtrancl_o @ R ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ Y3 ) @ R ) ) ) ).
% tranclD2
thf(fact_511_trancl__into__rtrancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_512_rtrancl__eq__or__trancl,axiom,
! [X4: $o,Y3: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_rtrancl_o @ R ) )
= ( ( X4 = Y3 )
| ( ( X4 != Y3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_513_rtrancl__into__trancl1,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_514_rtrancl__into__trancl2,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_515_rtrancl__trancl__trancl,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,Z: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z ) @ ( transitive_trancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_516_trancl__rtrancl__trancl,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( transitive_trancl_o @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_517_trancl__induct2,axiom,
! [Ax: relational_fmla_a_b,Ay: set_Pr1261947904930325089at_nat,Bx: relational_fmla_a_b,By: set_Pr1261947904930325089at_nat,R3: set_Pr3205448351804440111at_nat,P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ Bx @ By ) ) @ ( transi5629900152392516758at_nat @ R3 ) )
=> ( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ A2 @ B2 ) ) @ R3 )
=> ( P2 @ A2 @ B2 ) )
=> ( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat,Aa: relational_fmla_a_b,Ba: set_Pr1261947904930325089at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ Ax @ Ay ) @ ( produc760948067230524457at_nat @ A2 @ B2 ) ) @ ( transi5629900152392516758at_nat @ R3 ) )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ ( produc760948067230524457at_nat @ A2 @ B2 ) @ ( produc760948067230524457at_nat @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A2 @ B2 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_518_trancl__induct2,axiom,
! [Ax: $o,Ay: $o,Bx: $o,By: $o,R3: set_Pr7011910401827753159od_o_o,P2: $o > $o > $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ Bx @ By ) ) @ ( transi1494748803203666376od_o_o @ R3 ) )
=> ( ! [A2: $o,B2: $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ A2 @ B2 ) ) @ R3 )
=> ( P2 @ A2 @ B2 ) )
=> ( ! [A2: $o,B2: $o,Aa: $o,Ba: $o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ Ax @ Ay ) @ ( product_Pair_o_o @ A2 @ B2 ) ) @ ( transi1494748803203666376od_o_o @ R3 ) )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ ( product_Pair_o_o @ Aa @ Ba ) ) @ R3 )
=> ( ( P2 @ A2 @ B2 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_519_rel__restrict__def,axiom,
( rel_re178346425959758229at_nat
= ( ^ [R4: set_Pr3205448351804440111at_nat,A7: set_Pr2645174627780777389at_nat] :
( collec2298268480447286202at_nat
@ ( produc8549805575779043836_nat_o
@ ^ [V2: produc5825016348098550007at_nat,W2: produc5825016348098550007at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ V2 @ W2 ) @ R4 )
& ~ ( member6956540099943067662at_nat @ V2 @ A7 )
& ~ ( member6956540099943067662at_nat @ W2 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_520_rel__restrict__def,axiom,
( rel_re6977384794829015305od_o_o
= ( ^ [R4: set_Pr7011910401827753159od_o_o,A7: set_Product_prod_o_o] :
( collec7923239488154759506od_o_o
@ ( produc7860636428938663212_o_o_o
@ ^ [V2: product_prod_o_o,W2: product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ V2 @ W2 ) @ R4 )
& ~ ( member7466972457876170832od_o_o @ V2 @ A7 )
& ~ ( member7466972457876170832od_o_o @ W2 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_521_rel__restrict__def,axiom,
( rel_restrict_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat,A7: set_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [V2: nat,W2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V2 @ W2 ) @ R4 )
& ~ ( member_nat @ V2 @ A7 )
& ~ ( member_nat @ W2 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_522_rel__restrict__def,axiom,
( rel_restrict_o
= ( ^ [R4: set_Product_prod_o_o,A7: set_o] :
( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [V2: $o,W2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V2 @ W2 ) @ R4 )
& ~ ( member_o @ V2 @ A7 )
& ~ ( member_o @ W2 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_523_max__ext__eq,axiom,
( max_ext_o
= ( ^ [R4: set_Product_prod_o_o] :
( collec1488362387706010770_set_o
@ ( produc8987465890351537004et_o_o
@ ^ [X8: set_o,Y9: set_o] :
( ( finite_finite_o @ X8 )
& ( finite_finite_o @ Y9 )
& ( Y9 != bot_bot_set_o )
& ! [X: $o] :
( ( member_o @ X @ X8 )
=> ? [Y4: $o] :
( ( member_o @ Y4 @ Y9 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_524_max__ext__eq,axiom,
( max_ext_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
( collec6662362479098859352et_nat
@ ( produc6247414631856714078_nat_o
@ ^ [X8: set_nat,Y9: set_nat] :
( ( finite_finite_nat @ X8 )
& ( finite_finite_nat @ Y9 )
& ( Y9 != bot_bot_set_nat )
& ! [X: nat] :
( ( member_nat @ X @ X8 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ Y9 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_525_max__ext__eq,axiom,
( max_ex7572203241849361634od_o_o
= ( ^ [R4: set_Pr7011910401827753159od_o_o] :
( collec3302784863738620242od_o_o
@ ( produc9128050996712119596_o_o_o
@ ^ [X8: set_Product_prod_o_o,Y9: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ X8 )
& ( finite6120865539452801872od_o_o @ Y9 )
& ( Y9 != bot_bo7073875226086086771od_o_o )
& ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ X8 )
=> ? [Y4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y4 @ Y9 )
& ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_526_max__ext__eq,axiom,
( max_ex5394131240172625916at_nat
= ( ^ [R4: set_Pr3205448351804440111at_nat] :
( collec4440844842671817510at_nat
@ ( produc2686380336181240976_nat_o
@ ^ [X8: set_Pr2645174627780777389at_nat,Y9: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ X8 )
& ( finite6790245451575510286at_nat @ Y9 )
& ( Y9 != bot_bo1973934853101755969at_nat )
& ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ X8 )
=> ? [Y4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y4 @ Y9 )
& ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X @ Y4 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_527_trancl__single,axiom,
! [A: $o,B: $o] :
( ( transitive_trancl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ bot_bo7073875226086086771od_o_o ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ bot_bo7073875226086086771od_o_o ) ) ).
% trancl_single
thf(fact_528_trancl__subset__Sigma__aux,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,R3: set_Pr3205448351804440111at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ( ( ord_le559745420748760527at_nat @ R3
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : A4 ) )
=> ( ( A = B )
| ( member6956540099943067662at_nat @ A @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_529_trancl__subset__Sigma__aux,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A4: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ( ( ord_le6026832634849756263od_o_o @ R3
@ ( produc6514336722952383752od_o_o @ A4
@ ^ [Uu: product_prod_o_o] : A4 ) )
=> ( ( A = B )
| ( member7466972457876170832od_o_o @ A @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_530_trancl__subset__Sigma__aux,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( ord_le3146513528884898305at_nat @ R3
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) )
=> ( ( A = B )
| ( member_nat @ A @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_531_trancl__subset__Sigma__aux,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,A4: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( ( ord_le192195778054999463od_o_o @ R3
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : A4 ) )
=> ( ( A = B )
| ( member_o @ A @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_532_symclp__symcl__eq,axiom,
! [R3: set_Product_prod_o_o] :
( ( transitive_symclp_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R3 ) )
= ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ ( symcl_o @ R3 ) ) ) ) ).
% symclp_symcl_eq
thf(fact_533_prod__rel__def__internal,axiom,
( prod_r2259558363451189650at_nat
= ( ^ [R1: set_Pr2645174627780777389at_nat,R22: set_Pr2645174627780777389at_nat] :
( collec8707217689086466714at_nat
@ ( produc7572467186678597092_nat_o
@ ( produc2965676052996423234_nat_o
@ ^ [A3: relational_fmla_a_b,B3: relational_fmla_a_b] :
( produc410239310623530412_nat_o
@ ^ [A8: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A3 @ A8 ) @ R1 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B3 @ B8 ) @ R22 ) ) ) ) ) ) ) ) ).
% prod_rel_def_internal
thf(fact_534_prod__rel__def__internal,axiom,
( prod_r2099616133524126062at_o_o
= ( ^ [R1: set_Pr2645174627780777389at_nat,R22: set_Product_prod_o_o] :
( collec4594267173528951610_nat_o
@ ( produc6767099494957451476at_o_o
@ ( produc8009524868414171628at_o_o
@ ^ [A3: relational_fmla_a_b,B3: $o] :
( produc261481253472176821at_o_o
@ ^ [A8: set_Pr1261947904930325089at_nat,B8: $o] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A3 @ A8 ) @ R1 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ B8 ) @ R22 ) ) ) ) ) ) ) ) ).
% prod_rel_def_internal
thf(fact_535_prod__rel__def__internal,axiom,
( prod_r7822463253123016942at_nat
= ( ^ [R1: set_Product_prod_o_o,R22: set_Pr2645174627780777389at_nat] :
( collec2370594742830584660at_nat
@ ( produc3592705271764145466_nat_o
@ ( produc651025734141659802_nat_o
@ ^ [A3: $o,B3: relational_fmla_a_b] :
( produc3241507078245823331_nat_o
@ ^ [A8: $o,B8: set_Pr1261947904930325089at_nat] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A8 ) @ R1 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B3 @ B8 ) @ R22 ) ) ) ) ) ) ) ) ).
% prod_rel_def_internal
thf(fact_536_prod__rel__def__internal,axiom,
( prod_rel_o_o_o_o
= ( ^ [R1: set_Product_prod_o_o,R22: set_Product_prod_o_o] :
( collec7923239488154759506od_o_o
@ ( produc7860636428938663212_o_o_o
@ ( produc5911696763256879298_o_o_o
@ ^ [A3: $o,B3: $o] :
( produc6197397395684419436_o_o_o
@ ^ [A8: $o,B8: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A8 ) @ R1 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ B8 ) @ R22 ) ) ) ) ) ) ) ) ).
% prod_rel_def_internal
thf(fact_537_subsetI,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( member6956540099943067662at_nat @ X3 @ B5 ) )
=> ( ord_le8520675249591772685at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_538_subsetI,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( member7466972457876170832od_o_o @ X3 @ B5 ) )
=> ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ).
% subsetI
thf(fact_539_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_540_insert__iff,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B @ A4 ) )
= ( ( A = B )
| ( member6956540099943067662at_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_541_insert__iff,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ A4 ) )
= ( ( A = B )
| ( member7466972457876170832od_o_o @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_542_insert__iff,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
= ( ( A = B )
| ( member_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_543_insertCI,axiom,
! [A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,B: produc5825016348098550007at_nat] :
( ( ~ ( member6956540099943067662at_nat @ A @ B5 )
=> ( A = B ) )
=> ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_544_insertCI,axiom,
! [A: product_prod_o_o,B5: set_Product_prod_o_o,B: product_prod_o_o] :
( ( ~ ( member7466972457876170832od_o_o @ A @ B5 )
=> ( A = B ) )
=> ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ B5 ) ) ) ).
% insertCI
thf(fact_545_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_546_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_547_empty__subsetI,axiom,
! [A4: set_Product_prod_o_o] : ( ord_le192195778054999463od_o_o @ bot_bo7073875226086086771od_o_o @ A4 ) ).
% empty_subsetI
thf(fact_548_empty__subsetI,axiom,
! [A4: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ bot_bo1973934853101755969at_nat @ A4 ) ).
% empty_subsetI
thf(fact_549_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_550_subset__empty,axiom,
! [A4: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ bot_bo7073875226086086771od_o_o )
= ( A4 = bot_bo7073875226086086771od_o_o ) ) ).
% subset_empty
thf(fact_551_subset__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ bot_bo1973934853101755969at_nat )
= ( A4 = bot_bo1973934853101755969at_nat ) ) ).
% subset_empty
thf(fact_552_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_553_singletonI,axiom,
! [A: product_prod_o_o] : ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singletonI
thf(fact_554_singletonI,axiom,
! [A: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singletonI
thf(fact_555_insert__subset,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ B5 )
= ( ( member6956540099943067662at_nat @ X4 @ B5 )
& ( ord_le8520675249591772685at_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_556_insert__subset,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ B5 )
= ( ( member7466972457876170832od_o_o @ X4 @ B5 )
& ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_557_insert__subset,axiom,
! [X4: nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A4 ) @ B5 )
= ( ( member_nat @ X4 @ B5 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_558_finite__insert,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( insert3260075854425521959at_nat @ A @ A4 ) )
= ( finite6790245451575510286at_nat @ A4 ) ) ).
% finite_insert
thf(fact_559_insert__Diff1,axiom,
! [X4: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X4 @ B5 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ B5 )
= ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_560_insert__Diff1,axiom,
! [X4: product_prod_o_o,B5: set_Product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ B5 )
=> ( ( minus_7052513330764642158od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ B5 )
= ( minus_7052513330764642158od_o_o @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_561_insert__Diff1,axiom,
! [X4: nat,B5: set_nat,A4: set_nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A4 ) @ B5 )
= ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_562_Diff__insert0,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ B5 ) )
= ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_563_Diff__insert0,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ B5 ) )
= ( minus_7052513330764642158od_o_o @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_564_Diff__insert0,axiom,
! [X4: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X4 @ A4 )
=> ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ B5 ) )
= ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_565_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y10: nat,Z4: nat] : ( Y10 = Z4 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_566_singleton__conv2,axiom,
! [A: product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ( ^ [Y10: product_prod_o_o,Z4: product_prod_o_o] : ( Y10 = Z4 )
@ A ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singleton_conv2
thf(fact_567_singleton__conv2,axiom,
! [A: produc5825016348098550007at_nat] :
( ( collec7697611494764367948at_nat
@ ( ^ [Y10: produc5825016348098550007at_nat,Z4: produc5825016348098550007at_nat] : ( Y10 = Z4 )
@ A ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singleton_conv2
thf(fact_568_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_569_singleton__conv,axiom,
! [A: product_prod_o_o] :
( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] : ( X = A ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).
% singleton_conv
thf(fact_570_singleton__conv,axiom,
! [A: produc5825016348098550007at_nat] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( X = A ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ).
% singleton_conv
thf(fact_571_singleton__insert__inj__eq_H,axiom,
! [A: nat,A4: set_nat,B: nat] :
( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_572_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ A @ A4 )
= ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
= ( ( A = B )
& ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_573_singleton__insert__inj__eq_H,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ A4 )
= ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) )
= ( ( A = B )
& ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_574_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A4: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_575_singleton__insert__inj__eq,axiom,
! [B: product_prod_o_o,A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o )
= ( insert6201435330877294327od_o_o @ A @ A4 ) )
= ( ( A = B )
& ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_576_singleton__insert__inj__eq,axiom,
! [B: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat )
= ( insert3260075854425521959at_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_577_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_578_Diff__eq__empty__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( minus_7052513330764642158od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o )
= ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_579_Diff__eq__empty__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( minus_6698950876951835142at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat )
= ( ord_le8520675249591772685at_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_580_insert__Diff__single,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_581_insert__Diff__single,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ A @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= ( insert6201435330877294327od_o_o @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_582_insert__Diff__single,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ A @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= ( insert3260075854425521959at_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_583_finite__Diff__insert,axiom,
! [A4: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) ) )
= ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff_insert
thf(fact_584_finite__Collect__subsets,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( finite2560455063271695812at_nat
@ ( collec5061078246732527874at_nat
@ ^ [B7: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ B7 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_585_subset__Collect__conv,axiom,
! [S: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ S @ ( collec7697611494764367948at_nat @ P2 ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ S )
=> ( P2 @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_586_mk__disjoint__insert,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ? [B9: set_Pr2645174627780777389at_nat] :
( ( A4
= ( insert3260075854425521959at_nat @ A @ B9 ) )
& ~ ( member6956540099943067662at_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_587_mk__disjoint__insert,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ? [B9: set_Product_prod_o_o] :
( ( A4
= ( insert6201435330877294327od_o_o @ A @ B9 ) )
& ~ ( member7466972457876170832od_o_o @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_588_mk__disjoint__insert,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ? [B9: set_nat] :
( ( A4
= ( insert_nat @ A @ B9 ) )
& ~ ( member_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_589_Collect__mono__iff,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ ( collec7697611494764367948at_nat @ P2 ) @ ( collec7697611494764367948at_nat @ Q ) )
= ( ! [X: produc5825016348098550007at_nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_590_subset__insert,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ B5 ) )
= ( ord_le8520675249591772685at_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_591_subset__insert,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ B5 ) )
= ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_592_subset__insert,axiom,
! [X4: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X4 @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_593_insert__eq__iff,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ A4 )
=> ( ~ ( member6956540099943067662at_nat @ B @ B5 )
=> ( ( ( insert3260075854425521959at_nat @ A @ A4 )
= ( insert3260075854425521959at_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C5: set_Pr2645174627780777389at_nat] :
( ( A4
= ( insert3260075854425521959at_nat @ B @ C5 ) )
& ~ ( member6956540099943067662at_nat @ B @ C5 )
& ( B5
= ( insert3260075854425521959at_nat @ A @ C5 ) )
& ~ ( member6956540099943067662at_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_594_insert__eq__iff,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ B @ B5 )
=> ( ( ( insert6201435330877294327od_o_o @ A @ A4 )
= ( insert6201435330877294327od_o_o @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C5: set_Product_prod_o_o] :
( ( A4
= ( insert6201435330877294327od_o_o @ B @ C5 ) )
& ~ ( member7466972457876170832od_o_o @ B @ C5 )
& ( B5
= ( insert6201435330877294327od_o_o @ A @ C5 ) )
& ~ ( member7466972457876170832od_o_o @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_595_insert__eq__iff,axiom,
! [A: nat,A4: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A4 = B5 ) )
& ( ( A != B )
=> ? [C5: set_nat] :
( ( A4
= ( insert_nat @ B @ C5 ) )
& ~ ( member_nat @ B @ C5 )
& ( B5
= ( insert_nat @ A @ C5 ) )
& ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_596_insert__absorb,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( insert3260075854425521959at_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_597_insert__absorb,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( insert6201435330877294327od_o_o @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_598_insert__absorb,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_599_insert__ident,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ~ ( member6956540099943067662at_nat @ X4 @ B5 )
=> ( ( ( insert3260075854425521959at_nat @ X4 @ A4 )
= ( insert3260075854425521959at_nat @ X4 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_600_insert__ident,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ X4 @ B5 )
=> ( ( ( insert6201435330877294327od_o_o @ X4 @ A4 )
= ( insert6201435330877294327od_o_o @ X4 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_601_insert__ident,axiom,
! [X4: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ X4 @ A4 )
=> ( ~ ( member_nat @ X4 @ B5 )
=> ( ( ( insert_nat @ X4 @ A4 )
= ( insert_nat @ X4 @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_602_Collect__mono,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ! [X3: produc5825016348098550007at_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8520675249591772685at_nat @ ( collec7697611494764367948at_nat @ P2 ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_603_subset__iff,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
! [T2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ T2 @ A7 )
=> ( member6956540099943067662at_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_604_subset__iff,axiom,
( ord_le192195778054999463od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
! [T2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ T2 @ A7 )
=> ( member7466972457876170832od_o_o @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_605_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A7 )
=> ( member_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_606_Set_Oset__insert,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X4 @ A4 )
=> ~ ! [B9: set_Pr2645174627780777389at_nat] :
( ( A4
= ( insert3260075854425521959at_nat @ X4 @ B9 ) )
=> ( member6956540099943067662at_nat @ X4 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_607_Set_Oset__insert,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ~ ! [B9: set_Product_prod_o_o] :
( ( A4
= ( insert6201435330877294327od_o_o @ X4 @ B9 ) )
=> ( member7466972457876170832od_o_o @ X4 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_608_Set_Oset__insert,axiom,
! [X4: nat,A4: set_nat] :
( ( member_nat @ X4 @ A4 )
=> ~ ! [B9: set_nat] :
( ( A4
= ( insert_nat @ X4 @ B9 ) )
=> ( member_nat @ X4 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_609_subset__eq,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A7 )
=> ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_610_subset__eq,axiom,
( ord_le192195778054999463od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A7 )
=> ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_611_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A7 )
=> ( member_nat @ X @ B7 ) ) ) ) ).
% subset_eq
thf(fact_612_insertI2,axiom,
! [A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,B: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A @ B5 )
=> ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_613_insertI2,axiom,
! [A: product_prod_o_o,B5: set_Product_prod_o_o,B: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ B5 )
=> ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ B5 ) ) ) ).
% insertI2
thf(fact_614_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_615_insertI1,axiom,
! [A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] : ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_616_insertI1,axiom,
! [A: product_prod_o_o,B5: set_Product_prod_o_o] : ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ A @ B5 ) ) ).
% insertI1
thf(fact_617_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_618_subsetD,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,C: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( ( member6956540099943067662at_nat @ C @ A4 )
=> ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_619_subsetD,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,C: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
=> ( ( member7466972457876170832od_o_o @ C @ A4 )
=> ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% subsetD
thf(fact_620_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_621_insertE,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ ( insert3260075854425521959at_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member6956540099943067662at_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_622_insertE,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ A4 ) )
=> ( ( A != B )
=> ( member7466972457876170832od_o_o @ A @ A4 ) ) ) ).
% insertE
thf(fact_623_insertE,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_624_in__mono,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( member6956540099943067662at_nat @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_625_in__mono,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
=> ( ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( member7466972457876170832od_o_o @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_626_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_627_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_628_subset__singleton__iff,axiom,
! [X5: set_Product_prod_o_o,A: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ X5 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( ( X5 = bot_bo7073875226086086771od_o_o )
| ( X5
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_629_subset__singleton__iff,axiom,
! [X5: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ X5 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( ( X5 = bot_bo1973934853101755969at_nat )
| ( X5
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_630_subset__singletonD,axiom,
! [A4: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A4 = bot_bot_set_nat )
| ( A4
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_631_subset__singletonD,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
=> ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( A4
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% subset_singletonD
thf(fact_632_subset__singletonD,axiom,
! [A4: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
=> ( ( A4 = bot_bo1973934853101755969at_nat )
| ( A4
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_633_Collect__subset,axiom,
! [A4: set_Product_prod_o_o,P2: product_prod_o_o > $o] :
( ord_le192195778054999463od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
& ( P2 @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_634_Collect__subset,axiom,
! [A4: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( P2 @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_635_Collect__subset,axiom,
! [A4: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o] :
( ord_le8520675249591772685at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
& ( P2 @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_636_insert__Collect,axiom,
! [A: produc5825016348098550007at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( insert3260075854425521959at_nat @ A @ ( collec7697611494764367948at_nat @ P2 ) )
= ( collec7697611494764367948at_nat
@ ^ [U2: produc5825016348098550007at_nat] :
( ( U2 != A )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_637_insert__compr,axiom,
( insert6201435330877294327od_o_o
= ( ^ [A3: product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( X = A3 )
| ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_638_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B7: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
| ( member_nat @ X @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_639_insert__compr,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A3 )
| ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_640_subset__Diff__insert,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat,C2: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ ( minus_6698950876951835142at_nat @ B5 @ ( insert3260075854425521959at_nat @ X4 @ C2 ) ) )
= ( ( ord_le8520675249591772685at_nat @ A4 @ ( minus_6698950876951835142at_nat @ B5 @ C2 ) )
& ~ ( member6956540099943067662at_nat @ X4 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_641_subset__Diff__insert,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,X4: product_prod_o_o,C2: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ ( minus_7052513330764642158od_o_o @ B5 @ ( insert6201435330877294327od_o_o @ X4 @ C2 ) ) )
= ( ( ord_le192195778054999463od_o_o @ A4 @ ( minus_7052513330764642158od_o_o @ B5 @ C2 ) )
& ~ ( member7466972457876170832od_o_o @ X4 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_642_subset__Diff__insert,axiom,
! [A4: set_nat,B5: set_nat,X4: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ C2 ) )
& ~ ( member_nat @ X4 @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_643_finite__subset__induct_H,axiom,
! [F4: set_nat,A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A2 @ A4 )
=> ( ( ord_less_eq_set_nat @ F5 @ A4 )
=> ( ~ ( member_nat @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert_nat @ A2 @ F5 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_644_finite__subset__induct_H,axiom,
! [F4: set_Product_prod_o_o,A4: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ F4 )
=> ( ( ord_le192195778054999463od_o_o @ F4 @ A4 )
=> ( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ! [A2: product_prod_o_o,F5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ F5 )
=> ( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( ord_le192195778054999463od_o_o @ F5 @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert6201435330877294327od_o_o @ A2 @ F5 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_645_finite__subset__induct_H,axiom,
! [F4: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ F4 )
=> ( ( ord_le8520675249591772685at_nat @ F4 @ A4 )
=> ( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ! [A2: produc5825016348098550007at_nat,F5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F5 )
=> ( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( ord_le8520675249591772685at_nat @ F5 @ A4 )
=> ( ~ ( member6956540099943067662at_nat @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert3260075854425521959at_nat @ A2 @ F5 ) ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_646_finite__subset__induct,axiom,
! [F4: set_nat,A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A2 @ A4 )
=> ( ~ ( member_nat @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert_nat @ A2 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_647_finite__subset__induct,axiom,
! [F4: set_Product_prod_o_o,A4: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ F4 )
=> ( ( ord_le192195778054999463od_o_o @ F4 @ A4 )
=> ( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ! [A2: product_prod_o_o,F5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ F5 )
=> ( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert6201435330877294327od_o_o @ A2 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_648_finite__subset__induct,axiom,
! [F4: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ F4 )
=> ( ( ord_le8520675249591772685at_nat @ F4 @ A4 )
=> ( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ! [A2: produc5825016348098550007at_nat,F5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F5 )
=> ( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ~ ( member6956540099943067662at_nat @ A2 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert3260075854425521959at_nat @ A2 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_649_Diff__single__insert,axiom,
! [A4: set_nat,X4: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X4 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_650_Diff__single__insert,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) @ B5 )
=> ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_651_Diff__single__insert,axiom,
! [A4: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) @ B5 )
=> ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_652_subset__insert__iff,axiom,
! [A4: set_nat,X4: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X4 @ B5 ) )
= ( ( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_653_subset__insert__iff,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ B5 ) )
= ( ( ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ord_le192195778054999463od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) @ B5 ) )
& ( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_654_subset__insert__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ B5 ) )
= ( ( ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ord_le8520675249591772685at_nat @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) @ B5 ) )
& ( ~ ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ord_le8520675249591772685at_nat @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_655_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_656_singleton__inject,axiom,
! [A: product_prod_o_o,B: product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o )
= ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_657_singleton__inject,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat )
= ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_658_insert__not__empty,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_659_insert__not__empty,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( insert6201435330877294327od_o_o @ A @ A4 )
!= bot_bo7073875226086086771od_o_o ) ).
% insert_not_empty
thf(fact_660_insert__not__empty,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( insert3260075854425521959at_nat @ A @ A4 )
!= bot_bo1973934853101755969at_nat ) ).
% insert_not_empty
thf(fact_661_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_662_doubleton__eq__iff,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,C: product_prod_o_o,D2: product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
= ( insert6201435330877294327od_o_o @ C @ ( insert6201435330877294327od_o_o @ D2 @ bot_bo7073875226086086771od_o_o ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_663_doubleton__eq__iff,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,C: produc5825016348098550007at_nat,D2: produc5825016348098550007at_nat] :
( ( ( insert3260075854425521959at_nat @ A @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) )
= ( insert3260075854425521959at_nat @ C @ ( insert3260075854425521959at_nat @ D2 @ bot_bo1973934853101755969at_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_664_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_665_singleton__iff,axiom,
! [B: product_prod_o_o,A: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_666_singleton__iff,axiom,
! [B: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ B @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_667_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_668_singletonD,axiom,
! [B: product_prod_o_o,A: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_669_singletonD,axiom,
! [B: produc5825016348098550007at_nat,A: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ B @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_670_finite_OinsertI,axiom,
! [A4: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( finite6790245451575510286at_nat @ ( insert3260075854425521959at_nat @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_671_subrelI,axiom,
! [R3: set_Pr2645174627780777389at_nat,S4: set_Pr2645174627780777389at_nat] :
( ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X3 @ Y ) @ R3 )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X3 @ Y ) @ S4 ) )
=> ( ord_le8520675249591772685at_nat @ R3 @ S4 ) ) ).
% subrelI
thf(fact_672_subrelI,axiom,
! [R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ! [X3: $o,Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ S4 ) )
=> ( ord_le192195778054999463od_o_o @ R3 @ S4 ) ) ).
% subrelI
thf(fact_673_insert__Diff__if,axiom,
! [X4: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ X4 @ B5 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ B5 )
= ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) )
& ( ~ ( member6956540099943067662at_nat @ X4 @ B5 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ B5 )
= ( insert3260075854425521959at_nat @ X4 @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_674_insert__Diff__if,axiom,
! [X4: product_prod_o_o,B5: set_Product_prod_o_o,A4: set_Product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ X4 @ B5 )
=> ( ( minus_7052513330764642158od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ B5 )
= ( minus_7052513330764642158od_o_o @ A4 @ B5 ) ) )
& ( ~ ( member7466972457876170832od_o_o @ X4 @ B5 )
=> ( ( minus_7052513330764642158od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ B5 )
= ( insert6201435330877294327od_o_o @ X4 @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_675_insert__Diff__if,axiom,
! [X4: nat,B5: set_nat,A4: set_nat] :
( ( ( member_nat @ X4 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A4 ) @ B5 )
= ( minus_minus_set_nat @ A4 @ B5 ) ) )
& ( ~ ( member_nat @ X4 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A4 ) @ B5 )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_676_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_677_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A @ bot_bo7073875226086086771od_o_o )
=> ( A = bot_bo7073875226086086771od_o_o ) ) ).
% bot.extremum_uniqueI
thf(fact_678_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A @ bot_bo1973934853101755969at_nat )
=> ( A = bot_bo1973934853101755969at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_679_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_680_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A @ bot_bo7073875226086086771od_o_o )
= ( A = bot_bo7073875226086086771od_o_o ) ) ).
% bot.extremum_unique
thf(fact_681_bot_Oextremum__unique,axiom,
! [A: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A @ bot_bo1973934853101755969at_nat )
= ( A = bot_bo1973934853101755969at_nat ) ) ).
% bot.extremum_unique
thf(fact_682_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_683_bot_Oextremum,axiom,
! [A: set_Product_prod_o_o] : ( ord_le192195778054999463od_o_o @ bot_bo7073875226086086771od_o_o @ A ) ).
% bot.extremum
thf(fact_684_bot_Oextremum,axiom,
! [A: set_Pr2645174627780777389at_nat] : ( ord_le8520675249591772685at_nat @ bot_bo1973934853101755969at_nat @ A ) ).
% bot.extremum
thf(fact_685_finite__has__minimal2,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ? [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
& ( ord_le2636248316962427079od_o_o @ X3 @ A )
& ! [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ A4 )
=> ( ( ord_le2636248316962427079od_o_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_686_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_687_finite__has__maximal2,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ? [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
& ( ord_le2636248316962427079od_o_o @ A @ X3 )
& ! [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ A4 )
=> ( ( ord_le2636248316962427079od_o_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_688_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_689_rev__finite__subset,axiom,
! [B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ B5 )
=> ( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_690_infinite__super,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ S @ T )
=> ( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ T ) ) ) ).
% infinite_super
thf(fact_691_finite__subset,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_692_times__subset__iff,axiom,
! [A4: set_nat,C2: set_nat,B5: set_nat,D: set_nat] :
( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C2 = bot_bot_set_nat )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_693_times__subset__iff,axiom,
! [A4: set_nat,C2: set_Product_prod_o_o,B5: set_nat,D: set_Product_prod_o_o] :
( ( ord_le6246349990275258466od_o_o
@ ( produc7535496041681465101od_o_o @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc7535496041681465101od_o_o @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C2 = bot_bo7073875226086086771od_o_o )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_le192195778054999463od_o_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_694_times__subset__iff,axiom,
! [A4: set_nat,C2: set_Pr2645174627780777389at_nat,B5: set_nat,D: set_Pr2645174627780777389at_nat] :
( ( ord_le2120664821234635944at_nat
@ ( produc5885360257917011089at_nat @ A4
@ ^ [Uu: nat] : C2 )
@ ( produc5885360257917011089at_nat @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C2 = bot_bo1973934853101755969at_nat )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_le8520675249591772685at_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_695_times__subset__iff,axiom,
! [A4: set_Product_prod_o_o,C2: set_nat,B5: set_Product_prod_o_o,D: set_nat] :
( ( ord_le9042521934239567548_o_nat
@ ( produc2027885580938973615_o_nat @ A4
@ ^ [Uu: product_prod_o_o] : C2 )
@ ( produc2027885580938973615_o_nat @ B5
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( C2 = bot_bot_set_nat )
| ( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_696_times__subset__iff,axiom,
! [A4: set_Product_prod_o_o,C2: set_Product_prod_o_o,B5: set_Product_prod_o_o,D: set_Product_prod_o_o] :
( ( ord_le6026832634849756263od_o_o
@ ( produc6514336722952383752od_o_o @ A4
@ ^ [Uu: product_prod_o_o] : C2 )
@ ( produc6514336722952383752od_o_o @ B5
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( C2 = bot_bo7073875226086086771od_o_o )
| ( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
& ( ord_le192195778054999463od_o_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_697_times__subset__iff,axiom,
! [A4: set_Product_prod_o_o,C2: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o,D: set_Pr2645174627780777389at_nat] :
( ( ord_le2889795178612012515at_nat
@ ( produc8851762376515582294at_nat @ A4
@ ^ [Uu: product_prod_o_o] : C2 )
@ ( produc8851762376515582294at_nat @ B5
@ ^ [Uu: product_prod_o_o] : D ) )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
| ( C2 = bot_bo1973934853101755969at_nat )
| ( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
& ( ord_le8520675249591772685at_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_698_times__subset__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,C2: set_nat,B5: set_Pr2645174627780777389at_nat,D: set_nat] :
( ( ord_le7648075807827627176at_nat
@ ( produc3607307171264959249at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : C2 )
@ ( produc3607307171264959249at_nat @ B5
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( C2 = bot_bot_set_nat )
| ( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_699_times__subset__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,C2: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat,D: set_Product_prod_o_o] :
( ( ord_le4930721854091266427od_o_o
@ ( produc7724249792059183142od_o_o @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : C2 )
@ ( produc7724249792059183142od_o_o @ B5
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( C2 = bot_bo7073875226086086771od_o_o )
| ( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
& ( ord_le192195778054999463od_o_o @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_700_times__subset__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,C2: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,D: set_Pr2645174627780777389at_nat] :
( ( ord_le559745420748760527at_nat
@ ( produc984725824077784504at_nat @ A4
@ ^ [Uu: produc5825016348098550007at_nat] : C2 )
@ ( produc984725824077784504at_nat @ B5
@ ^ [Uu: produc5825016348098550007at_nat] : D ) )
= ( ( A4 = bot_bo1973934853101755969at_nat )
| ( C2 = bot_bo1973934853101755969at_nat )
| ( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
& ( ord_le8520675249591772685at_nat @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_701_in__rtrancl__insert,axiom,
! [X4: product_prod_o_o,R: set_Product_prod_o_o,R3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ ( transitive_rtrancl_o @ R ) )
=> ( member7466972457876170832od_o_o @ X4 @ ( transitive_rtrancl_o @ ( insert6201435330877294327od_o_o @ R3 @ R ) ) ) ) ).
% in_rtrancl_insert
thf(fact_702_less__by__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( A4 = bot_bo7073875226086086771od_o_o )
=> ( ord_le192195778054999463od_o_o @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_703_rtrancl__mono__mp,axiom,
! [U3: set_Product_prod_o_o,V3: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ U3 @ V3 )
=> ( ( member7466972457876170832od_o_o @ X4 @ ( transitive_rtrancl_o @ U3 ) )
=> ( member7466972457876170832od_o_o @ X4 @ ( transitive_rtrancl_o @ V3 ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_704_trancl__mono__mp,axiom,
! [U3: set_Product_prod_o_o,V3: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ U3 @ V3 )
=> ( ( member7466972457876170832od_o_o @ X4 @ ( transitive_trancl_o @ U3 ) )
=> ( member7466972457876170832od_o_o @ X4 @ ( transitive_trancl_o @ V3 ) ) ) ) ).
% trancl_mono_mp
thf(fact_705_finite__remove__induct,axiom,
! [B5: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B5 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( A9 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A9 @ B5 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( P2 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_706_finite__remove__induct,axiom,
! [B5: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ B5 )
=> ( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ! [A9: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A9 )
=> ( ( A9 != bot_bo7073875226086086771od_o_o )
=> ( ( ord_le192195778054999463od_o_o @ A9 @ B5 )
=> ( ! [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A9 )
=> ( P2 @ ( minus_7052513330764642158od_o_o @ A9 @ ( insert6201435330877294327od_o_o @ X6 @ bot_bo7073875226086086771od_o_o ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_707_finite__remove__induct,axiom,
! [B5: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ B5 )
=> ( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ! [A9: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A9 )
=> ( ( A9 != bot_bo1973934853101755969at_nat )
=> ( ( ord_le8520675249591772685at_nat @ A9 @ B5 )
=> ( ! [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ A9 )
=> ( P2 @ ( minus_6698950876951835142at_nat @ A9 @ ( insert3260075854425521959at_nat @ X6 @ bot_bo1973934853101755969at_nat ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_708_remove__induct,axiom,
! [P2: set_nat > $o,B5: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( A9 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A9 @ B5 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( P2 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_709_remove__induct,axiom,
! [P2: set_Product_prod_o_o > $o,B5: set_Product_prod_o_o] :
( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ( ~ ( finite6120865539452801872od_o_o @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A9: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A9 )
=> ( ( A9 != bot_bo7073875226086086771od_o_o )
=> ( ( ord_le192195778054999463od_o_o @ A9 @ B5 )
=> ( ! [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A9 )
=> ( P2 @ ( minus_7052513330764642158od_o_o @ A9 @ ( insert6201435330877294327od_o_o @ X6 @ bot_bo7073875226086086771od_o_o ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_710_remove__induct,axiom,
! [P2: set_Pr2645174627780777389at_nat > $o,B5: set_Pr2645174627780777389at_nat] :
( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ( ~ ( finite6790245451575510286at_nat @ B5 )
=> ( P2 @ B5 ) )
=> ( ! [A9: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A9 )
=> ( ( A9 != bot_bo1973934853101755969at_nat )
=> ( ( ord_le8520675249591772685at_nat @ A9 @ B5 )
=> ( ! [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ A9 )
=> ( P2 @ ( minus_6698950876951835142at_nat @ A9 @ ( insert3260075854425521959at_nat @ X6 @ bot_bo1973934853101755969at_nat ) ) ) )
=> ( P2 @ A9 ) ) ) ) )
=> ( P2 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_711_Collect__conv__if2,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P2 @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_712_Collect__conv__if2,axiom,
! [P2: product_prod_o_o > $o,A: product_prod_o_o] :
( ( ( P2 @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( A = X )
& ( P2 @ X ) ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( A = X )
& ( P2 @ X ) ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_conv_if2
thf(fact_713_Collect__conv__if2,axiom,
! [P2: produc5825016348098550007at_nat > $o,A: produc5825016348098550007at_nat] :
( ( ( P2 @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( A = X )
& ( P2 @ X ) ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( A = X )
& ( P2 @ X ) ) )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_714_Collect__conv__if,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P2 @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_715_Collect__conv__if,axiom,
! [P2: product_prod_o_o > $o,A: product_prod_o_o] :
( ( ( P2 @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( X = A )
& ( P2 @ X ) ) )
= ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( X = A )
& ( P2 @ X ) ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% Collect_conv_if
thf(fact_716_Collect__conv__if,axiom,
! [P2: produc5825016348098550007at_nat > $o,A: produc5825016348098550007at_nat] :
( ( ( P2 @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A )
& ( P2 @ X ) ) )
= ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( X = A )
& ( P2 @ X ) ) )
= bot_bo1973934853101755969at_nat ) ) ) ).
% Collect_conv_if
thf(fact_717_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A4: set_nat] :
( ! [A9: set_nat] :
( ~ ( finite_finite_nat @ A9 )
=> ( P2 @ A9 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert_nat @ X3 @ F5 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_718_infinite__finite__induct,axiom,
! [P2: set_Product_prod_o_o > $o,A4: set_Product_prod_o_o] :
( ! [A9: set_Product_prod_o_o] :
( ~ ( finite6120865539452801872od_o_o @ A9 )
=> ( P2 @ A9 ) )
=> ( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o,F5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ F5 )
=> ( ~ ( member7466972457876170832od_o_o @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert6201435330877294327od_o_o @ X3 @ F5 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_719_infinite__finite__induct,axiom,
! [P2: set_Pr2645174627780777389at_nat > $o,A4: set_Pr2645174627780777389at_nat] :
( ! [A9: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ A9 )
=> ( P2 @ A9 ) )
=> ( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ! [X3: produc5825016348098550007at_nat,F5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F5 )
=> ( ~ ( member6956540099943067662at_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert3260075854425521959at_nat @ X3 @ F5 ) ) ) ) )
=> ( P2 @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_720_finite__ne__induct,axiom,
! [F4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( F5 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert_nat @ X3 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_721_finite__ne__induct,axiom,
! [F4: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ F4 )
=> ( ( F4 != bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o] : ( P2 @ ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
=> ( ! [X3: product_prod_o_o,F5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ F5 )
=> ( ( F5 != bot_bo7073875226086086771od_o_o )
=> ( ~ ( member7466972457876170832od_o_o @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert6201435330877294327od_o_o @ X3 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_722_finite__ne__induct,axiom,
! [F4: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ F4 )
=> ( ( F4 != bot_bo1973934853101755969at_nat )
=> ( ! [X3: produc5825016348098550007at_nat] : ( P2 @ ( insert3260075854425521959at_nat @ X3 @ bot_bo1973934853101755969at_nat ) )
=> ( ! [X3: produc5825016348098550007at_nat,F5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F5 )
=> ( ( F5 != bot_bo1973934853101755969at_nat )
=> ( ~ ( member6956540099943067662at_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert3260075854425521959at_nat @ X3 @ F5 ) ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_723_finite__induct,axiom,
! [F4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert_nat @ X3 @ F5 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_724_finite__induct,axiom,
! [F4: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ F4 )
=> ( ( P2 @ bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o,F5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ F5 )
=> ( ~ ( member7466972457876170832od_o_o @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert6201435330877294327od_o_o @ X3 @ F5 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_725_finite__induct,axiom,
! [F4: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ F4 )
=> ( ( P2 @ bot_bo1973934853101755969at_nat )
=> ( ! [X3: produc5825016348098550007at_nat,F5: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F5 )
=> ( ~ ( member6956540099943067662at_nat @ X3 @ F5 )
=> ( ( P2 @ F5 )
=> ( P2 @ ( insert3260075854425521959at_nat @ X3 @ F5 ) ) ) ) )
=> ( P2 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_726_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A7: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_727_finite_Osimps,axiom,
( finite6120865539452801872od_o_o
= ( ^ [A3: set_Product_prod_o_o] :
( ( A3 = bot_bo7073875226086086771od_o_o )
| ? [A7: set_Product_prod_o_o,B3: product_prod_o_o] :
( ( A3
= ( insert6201435330877294327od_o_o @ B3 @ A7 ) )
& ( finite6120865539452801872od_o_o @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_728_finite_Osimps,axiom,
( finite6790245451575510286at_nat
= ( ^ [A3: set_Pr2645174627780777389at_nat] :
( ( A3 = bot_bo1973934853101755969at_nat )
| ? [A7: set_Pr2645174627780777389at_nat,B3: produc5825016348098550007at_nat] :
( ( A3
= ( insert3260075854425521959at_nat @ B3 @ A7 ) )
& ( finite6790245451575510286at_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_729_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A9: set_nat] :
( ? [A2: nat] :
( A
= ( insert_nat @ A2 @ A9 ) )
=> ~ ( finite_finite_nat @ A9 ) ) ) ) ).
% finite.cases
thf(fact_730_finite_Ocases,axiom,
! [A: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A )
=> ( ( A != bot_bo7073875226086086771od_o_o )
=> ~ ! [A9: set_Product_prod_o_o] :
( ? [A2: product_prod_o_o] :
( A
= ( insert6201435330877294327od_o_o @ A2 @ A9 ) )
=> ~ ( finite6120865539452801872od_o_o @ A9 ) ) ) ) ).
% finite.cases
thf(fact_731_finite_Ocases,axiom,
! [A: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A )
=> ( ( A != bot_bo1973934853101755969at_nat )
=> ~ ! [A9: set_Pr2645174627780777389at_nat] :
( ? [A2: produc5825016348098550007at_nat] :
( A
= ( insert3260075854425521959at_nat @ A2 @ A9 ) )
=> ~ ( finite6790245451575510286at_nat @ A9 ) ) ) ) ).
% finite.cases
thf(fact_732_insert__minus__eq,axiom,
! [X4: nat,Y3: nat,A4: set_nat] :
( ( X4 != Y3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A4 ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_733_insert__minus__eq,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( X4 != Y3 )
=> ( ( minus_7052513330764642158od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ ( insert6201435330877294327od_o_o @ Y3 @ bot_bo7073875226086086771od_o_o ) )
= ( insert6201435330877294327od_o_o @ X4 @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ Y3 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% insert_minus_eq
thf(fact_734_insert__minus__eq,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( X4 != Y3 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ ( insert3260075854425521959at_nat @ Y3 @ bot_bo1973934853101755969at_nat ) )
= ( insert3260075854425521959at_nat @ X4 @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ Y3 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_735_set__minus__singleton__eq,axiom,
! [X4: nat,X5: set_nat] :
( ~ ( member_nat @ X4 @ X5 )
=> ( ( minus_minus_set_nat @ X5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_736_set__minus__singleton__eq,axiom,
! [X4: product_prod_o_o,X5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ X5 )
=> ( ( minus_7052513330764642158od_o_o @ X5 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_737_set__minus__singleton__eq,axiom,
! [X4: produc5825016348098550007at_nat,X5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ X5 )
=> ( ( minus_6698950876951835142at_nat @ X5 @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_738_Diff__insert__absorb,axiom,
! [X4: nat,A4: set_nat] :
( ~ ( member_nat @ X4 @ A4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A4 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_739_Diff__insert__absorb,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( minus_7052513330764642158od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_740_Diff__insert__absorb,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ X4 @ A4 )
=> ( ( minus_6698950876951835142at_nat @ ( insert3260075854425521959at_nat @ X4 @ A4 ) @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_741_Diff__insert2,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_742_Diff__insert2,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o,B5: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( minus_7052513330764642158od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_743_Diff__insert2,axiom,
! [A4: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_744_insert__Diff,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_745_insert__Diff,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( insert6201435330877294327od_o_o @ A @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_746_insert__Diff,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( insert3260075854425521959at_nat @ A @ ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_747_Diff__insert,axiom,
! [A4: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_748_Diff__insert,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o,B5: set_Product_prod_o_o] :
( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( minus_7052513330764642158od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ B5 ) @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ) ).
% Diff_insert
thf(fact_749_Diff__insert,axiom,
! [A4: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( minus_6698950876951835142at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( minus_6698950876951835142at_nat @ ( minus_6698950876951835142at_nat @ A4 @ B5 ) @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ) ).
% Diff_insert
thf(fact_750_insert__remove__id,axiom,
! [X4: nat,X5: set_nat] :
( ( member_nat @ X4 @ X5 )
=> ( X5
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ X5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_751_insert__remove__id,axiom,
! [X4: product_prod_o_o,X5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ X5 )
=> ( X5
= ( insert6201435330877294327od_o_o @ X4 @ ( minus_7052513330764642158od_o_o @ X5 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% insert_remove_id
thf(fact_752_insert__remove__id,axiom,
! [X4: produc5825016348098550007at_nat,X5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X4 @ X5 )
=> ( X5
= ( insert3260075854425521959at_nat @ X4 @ ( minus_6698950876951835142at_nat @ X5 @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_753_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_754_finite__has__minimal,axiom,
! [A4: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ? [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
& ! [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ A4 )
=> ( ( ord_le2636248316962427079od_o_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_755_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_756_finite__has__maximal,axiom,
! [A4: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ? [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
& ! [Xa: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xa @ A4 )
=> ( ( ord_le2636248316962427079od_o_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_757_subset__minus__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ).
% subset_minus_empty
thf(fact_758_subset__minus__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
=> ( ( minus_7052513330764642158od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o ) ) ).
% subset_minus_empty
thf(fact_759_subset__minus__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( ( minus_6698950876951835142at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat ) ) ).
% subset_minus_empty
thf(fact_760_symcl__insert,axiom,
! [X4: $o,Y3: $o,Qeq2: set_Product_prod_o_o] :
( ( symcl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ Qeq2 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ Y3 @ X4 ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( symcl_o @ Qeq2 ) ) ) ) ).
% symcl_insert
thf(fact_761_finite__empty__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A2: nat,A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( member_nat @ A2 @ A9 )
=> ( ( P2 @ A9 )
=> ( P2 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_762_finite__empty__induct,axiom,
! [A4: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A2: product_prod_o_o,A9: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A9 )
=> ( ( member7466972457876170832od_o_o @ A2 @ A9 )
=> ( ( P2 @ A9 )
=> ( P2 @ ( minus_7052513330764642158od_o_o @ A9 @ ( insert6201435330877294327od_o_o @ A2 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) )
=> ( P2 @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% finite_empty_induct
thf(fact_763_finite__empty__induct,axiom,
! [A4: set_Pr2645174627780777389at_nat,P2: set_Pr2645174627780777389at_nat > $o] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( P2 @ A4 )
=> ( ! [A2: produc5825016348098550007at_nat,A9: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A9 )
=> ( ( member6956540099943067662at_nat @ A2 @ A9 )
=> ( ( P2 @ A9 )
=> ( P2 @ ( minus_6698950876951835142at_nat @ A9 @ ( insert3260075854425521959at_nat @ A2 @ bot_bo1973934853101755969at_nat ) ) ) ) ) )
=> ( P2 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_764_infinite__coinduct,axiom,
! [X5: set_nat > $o,A4: set_nat] :
( ( X5 @ A4 )
=> ( ! [A9: set_nat] :
( ( X5 @ A9 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A9 )
& ( ( X5 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_765_infinite__coinduct,axiom,
! [X5: set_Product_prod_o_o > $o,A4: set_Product_prod_o_o] :
( ( X5 @ A4 )
=> ( ! [A9: set_Product_prod_o_o] :
( ( X5 @ A9 )
=> ? [X6: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X6 @ A9 )
& ( ( X5 @ ( minus_7052513330764642158od_o_o @ A9 @ ( insert6201435330877294327od_o_o @ X6 @ bot_bo7073875226086086771od_o_o ) ) )
| ~ ( finite6120865539452801872od_o_o @ ( minus_7052513330764642158od_o_o @ A9 @ ( insert6201435330877294327od_o_o @ X6 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) )
=> ~ ( finite6120865539452801872od_o_o @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_766_infinite__coinduct,axiom,
! [X5: set_Pr2645174627780777389at_nat > $o,A4: set_Pr2645174627780777389at_nat] :
( ( X5 @ A4 )
=> ( ! [A9: set_Pr2645174627780777389at_nat] :
( ( X5 @ A9 )
=> ? [X6: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X6 @ A9 )
& ( ( X5 @ ( minus_6698950876951835142at_nat @ A9 @ ( insert3260075854425521959at_nat @ X6 @ bot_bo1973934853101755969at_nat ) ) )
| ~ ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ A9 @ ( insert3260075854425521959at_nat @ X6 @ bot_bo1973934853101755969at_nat ) ) ) ) ) )
=> ~ ( finite6790245451575510286at_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_767_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_768_infinite__remove,axiom,
! [S: set_Product_prod_o_o,A: product_prod_o_o] :
( ~ ( finite6120865539452801872od_o_o @ S )
=> ~ ( finite6120865539452801872od_o_o @ ( minus_7052513330764642158od_o_o @ S @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% infinite_remove
thf(fact_769_infinite__remove,axiom,
! [S: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat] :
( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ ( minus_6698950876951835142at_nat @ S @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% infinite_remove
thf(fact_770_finite__set__of__finite__funs,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,D2: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( finite6120865539452801872od_o_o @ B5 )
=> ( finite9004333502519276988od_o_o
@ ( collec5756396617468125178od_o_o
@ ^ [F3: product_prod_o_o > product_prod_o_o] :
! [X: product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ X @ A4 )
=> ( member7466972457876170832od_o_o @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member7466972457876170832od_o_o @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_771_finite__set__of__finite__funs,axiom,
! [A4: set_Product_prod_o_o,B5: set_nat,D2: nat] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite7641158932667466897_o_nat
@ ( collec6848040635926443731_o_nat
@ ^ [F3: product_prod_o_o > nat] :
! [X: product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ X @ A4 )
=> ( member_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member7466972457876170832od_o_o @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_772_finite__set__of__finite__funs,axiom,
! [A4: set_nat,B5: set_Product_prod_o_o,D2: product_prod_o_o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite6120865539452801872od_o_o @ B5 )
=> ( finite5919162904821268791od_o_o
@ ( collec5126044608080245625od_o_o
@ ^ [F3: nat > product_prod_o_o] :
! [X: nat] :
( ( ( member_nat @ X @ A4 )
=> ( member7466972457876170832od_o_o @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_773_finite__set__of__finite__funs,axiom,
! [A4: set_nat,B5: set_nat,D2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [F3: nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A4 )
=> ( member_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_774_finite__set__of__finite__funs,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat,D2: produc5825016348098550007at_nat] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( finite3406476425668988728at_nat
@ ( collec5327694418077629562at_nat
@ ^ [F3: product_prod_o_o > produc5825016348098550007at_nat] :
! [X: product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ X @ A4 )
=> ( member6956540099943067662at_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member7466972457876170832od_o_o @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_775_finite__set__of__finite__funs,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat,D2: produc5825016348098550007at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( finite503502276391589501at_nat
@ ( collec8011977876520774843at_nat
@ ^ [F3: nat > produc5825016348098550007at_nat] :
! [X: nat] :
( ( ( member_nat @ X @ A4 )
=> ( member6956540099943067662at_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_776_finite__set__of__finite__funs,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o,D2: product_prod_o_o] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite6120865539452801872od_o_o @ B5 )
=> ( finite4312028068632047312od_o_o
@ ( collec6233246061040688146od_o_o
@ ^ [F3: produc5825016348098550007at_nat > product_prod_o_o] :
! [X: produc5825016348098550007at_nat] :
( ( ( member6956540099943067662at_nat @ X @ A4 )
=> ( member7466972457876170832od_o_o @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member6956540099943067662at_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_777_finite__set__of__finite__funs,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat,D2: nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite7488119905747684989at_nat
@ ( collec5773223469022094523at_nat
@ ^ [F3: produc5825016348098550007at_nat > nat] :
! [X: produc5825016348098550007at_nat] :
( ( ( member6956540099943067662at_nat @ X @ A4 )
=> ( member_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member6956540099943067662at_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_778_finite__set__of__finite__funs,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,D2: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( finite6790245451575510286at_nat @ B5 )
=> ( finite5301788869328653092at_nat
@ ( collec1264455344780912226at_nat
@ ^ [F3: produc5825016348098550007at_nat > produc5825016348098550007at_nat] :
! [X: produc5825016348098550007at_nat] :
( ( ( member6956540099943067662at_nat @ X @ A4 )
=> ( member6956540099943067662at_nat @ ( F3 @ X ) @ B5 ) )
& ( ~ ( member6956540099943067662at_nat @ X @ A4 )
=> ( ( F3 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_779_trancl__over__edgeE,axiom,
! [U: $o,W: $o,V1: $o,V22: $o,E: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ W ) @ ( transitive_trancl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ V1 @ V22 ) @ E ) ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ W ) @ ( transitive_trancl_o @ E ) )
=> ~ ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V1 ) @ ( transitive_rtrancl_o @ E ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V22 @ W ) @ ( transitive_rtrancl_o @ E ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_780_eq__UNIV__iff,axiom,
! [S: set_Pr2645174627780777389at_nat] :
( ( S = top_to4412277702576798813at_nat )
= ( ! [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_781_eq__UNIV__iff,axiom,
! [S: set_Product_prod_o_o] :
( ( S = top_to7721136755696657239od_o_o )
= ( ! [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_782_eq__UNIV__iff,axiom,
! [S: set_nat] :
( ( S = top_top_set_nat )
= ( ! [X: nat] : ( member_nat @ X @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_783_it__step__insert__iff,axiom,
! [It: set_nat,S: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ It @ S )
=> ( ( member_nat @ X4 @ It )
=> ( ( minus_minus_set_nat @ S @ ( minus_minus_set_nat @ It @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_784_it__step__insert__iff,axiom,
! [It: set_Product_prod_o_o,S: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ It @ S )
=> ( ( member7466972457876170832od_o_o @ X4 @ It )
=> ( ( minus_7052513330764642158od_o_o @ S @ ( minus_7052513330764642158od_o_o @ It @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) )
= ( insert6201435330877294327od_o_o @ X4 @ ( minus_7052513330764642158od_o_o @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_785_it__step__insert__iff,axiom,
! [It: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( ord_le8520675249591772685at_nat @ It @ S )
=> ( ( member6956540099943067662at_nat @ X4 @ It )
=> ( ( minus_6698950876951835142at_nat @ S @ ( minus_6698950876951835142at_nat @ It @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) )
= ( insert3260075854425521959at_nat @ X4 @ ( minus_6698950876951835142at_nat @ S @ It ) ) ) ) ) ).
% it_step_insert_iff
thf(fact_786_case__prod__mono,axiom,
! [P: produc5825016348098550007at_nat,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,F2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc760948067230524457at_nat @ A2 @ B2 ) )
=> ( ord_less_eq_o @ ( F @ A2 @ B2 ) @ ( F2 @ A2 @ B2 ) ) )
=> ( ord_less_eq_o @ ( produc5861013030283631554_nat_o @ F @ P ) @ ( produc5861013030283631554_nat_o @ F2 @ P ) ) ) ).
% case_prod_mono
thf(fact_787_case__prod__mono,axiom,
! [P: product_prod_o_o,F: $o > $o > $o,F2: $o > $o > $o] :
( ! [A2: $o,B2: $o] :
( ( P
= ( product_Pair_o_o @ A2 @ B2 ) )
=> ( ord_less_eq_o @ ( F @ A2 @ B2 ) @ ( F2 @ A2 @ B2 ) ) )
=> ( ord_less_eq_o @ ( produc6197397395684419436_o_o_o @ F @ P ) @ ( produc6197397395684419436_o_o_o @ F2 @ P ) ) ) ).
% case_prod_mono
thf(fact_788_diff__shunt__var,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_789_diff__shunt__var,axiom,
! [X4: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
( ( ( minus_7052513330764642158od_o_o @ X4 @ Y3 )
= bot_bo7073875226086086771od_o_o )
= ( ord_le192195778054999463od_o_o @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_790_diff__shunt__var,axiom,
! [X4: set_Pr2645174627780777389at_nat,Y3: set_Pr2645174627780777389at_nat] :
( ( ( minus_6698950876951835142at_nat @ X4 @ Y3 )
= bot_bo1973934853101755969at_nat )
= ( ord_le8520675249591772685at_nat @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_791_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_792_the__elem__eq,axiom,
! [X4: product_prod_o_o] :
( ( the_el301066130825229084od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= X4 ) ).
% the_elem_eq
thf(fact_793_the__elem__eq,axiom,
! [X4: produc5825016348098550007at_nat] :
( ( the_el6542243382094817346at_nat @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_794_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_795_is__singletonI,axiom,
! [X4: product_prod_o_o] : ( is_sin2666361102579702811od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ).
% is_singletonI
thf(fact_796_is__singletonI,axiom,
! [X4: produc5825016348098550007at_nat] : ( is_sin7264216465531249411at_nat @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ).
% is_singletonI
thf(fact_797_pred__subset__eq,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( ord_le4411885398519701688_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ R )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ S ) )
= ( ord_le8520675249591772685at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_798_pred__subset__eq,axiom,
! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ord_le8356561246180588022_o_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ R )
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ S ) )
= ( ord_le192195778054999463od_o_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_799_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_800_less__eq__set__def,axiom,
( ord_le8520675249591772685at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( ord_le4411885398519701688_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A7 )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_801_less__eq__set__def,axiom,
( ord_le192195778054999463od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( ord_le8356561246180588022_o_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A7 )
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_802_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A7 )
@ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_803_Collect__case__prod__mono,axiom,
! [A4: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,B5: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o] :
( ( ord_le1013356598057680634_nat_o @ A4 @ B5 )
=> ( ord_le8520675249591772685at_nat @ ( collec7697611494764367948at_nat @ ( produc5861013030283631554_nat_o @ A4 ) ) @ ( collec7697611494764367948at_nat @ ( produc5861013030283631554_nat_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_804_Collect__case__prod__mono,axiom,
! [A4: $o > $o > $o,B5: $o > $o > $o] :
( ( ord_less_eq_o_o_o @ A4 @ B5 )
=> ( ord_le192195778054999463od_o_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ A4 ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_805_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
( A7
= ( insert_nat @ ( the_elem_nat @ A7 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_806_is__singleton__the__elem,axiom,
( is_sin2666361102579702811od_o_o
= ( ^ [A7: set_Product_prod_o_o] :
( A7
= ( insert6201435330877294327od_o_o @ ( the_el301066130825229084od_o_o @ A7 ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_807_is__singleton__the__elem,axiom,
( is_sin7264216465531249411at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat] :
( A7
= ( insert3260075854425521959at_nat @ ( the_el6542243382094817346at_nat @ A7 ) @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_808_pred__subset__eq2,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( ord_le1013356598057680634_nat_o
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ R )
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ S ) )
= ( ord_le8520675249591772685at_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_809_pred__subset__eq2,axiom,
! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ord_less_eq_o_o_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R )
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ S ) )
= ( ord_le192195778054999463od_o_o @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_810_is__singletonI_H,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_811_is__singletonI_H,axiom,
! [A4: set_Product_prod_o_o] :
( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o,Y: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( ( member7466972457876170832od_o_o @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_sin2666361102579702811od_o_o @ A4 ) ) ) ).
% is_singletonI'
thf(fact_812_is__singletonI_H,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( A4 != bot_bo1973934853101755969at_nat )
=> ( ! [X3: produc5825016348098550007at_nat,Y: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( ( member6956540099943067662at_nat @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_sin7264216465531249411at_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_813_is__singletonE,axiom,
! [A4: set_nat] :
( ( is_singleton_nat @ A4 )
=> ~ ! [X3: nat] :
( A4
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_814_is__singletonE,axiom,
! [A4: set_Product_prod_o_o] :
( ( is_sin2666361102579702811od_o_o @ A4 )
=> ~ ! [X3: product_prod_o_o] :
( A4
!= ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) ) ) ).
% is_singletonE
thf(fact_815_is__singletonE,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( is_sin7264216465531249411at_nat @ A4 )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( A4
!= ( insert3260075854425521959at_nat @ X3 @ bot_bo1973934853101755969at_nat ) ) ) ).
% is_singletonE
thf(fact_816_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
? [X: nat] :
( A7
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_817_is__singleton__def,axiom,
( is_sin2666361102579702811od_o_o
= ( ^ [A7: set_Product_prod_o_o] :
? [X: product_prod_o_o] :
( A7
= ( insert6201435330877294327od_o_o @ X @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% is_singleton_def
thf(fact_818_is__singleton__def,axiom,
( is_sin7264216465531249411at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat] :
? [X: produc5825016348098550007at_nat] :
( A7
= ( insert3260075854425521959at_nat @ X @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% is_singleton_def
thf(fact_819_linear__order__on__singleton,axiom,
! [X4: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X4 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ X4 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_820_linear__order__on__singleton,axiom,
! [X4: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_821_linear__order__on__singleton,axiom,
! [X4: product_prod_o_o] : ( order_7019990371155863132od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ X4 ) @ bot_bo2881837355931550515od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_822_linear__order__on__singleton,axiom,
! [X4: produc5825016348098550007at_nat] : ( order_3226272362920281346at_nat @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ X4 @ X4 ) @ bot_bo2147731989722242715at_nat ) ) ).
% linear_order_on_singleton
thf(fact_823_lnear__order__on__empty,axiom,
order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% lnear_order_on_empty
thf(fact_824_lnear__order__on__empty,axiom,
order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% lnear_order_on_empty
thf(fact_825_lnear__order__on__empty,axiom,
order_7019990371155863132od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo2881837355931550515od_o_o ).
% lnear_order_on_empty
thf(fact_826_lnear__order__on__empty,axiom,
order_3226272362920281346at_nat @ bot_bo1973934853101755969at_nat @ bot_bo2147731989722242715at_nat ).
% lnear_order_on_empty
thf(fact_827_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_828_subset__emptyI,axiom,
! [A4: set_Product_prod_o_o] :
( ! [X3: product_prod_o_o] :
~ ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( ord_le192195778054999463od_o_o @ A4 @ bot_bo7073875226086086771od_o_o ) ) ).
% subset_emptyI
thf(fact_829_subset__emptyI,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
~ ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( ord_le8520675249591772685at_nat @ A4 @ bot_bo1973934853101755969at_nat ) ) ).
% subset_emptyI
thf(fact_830_transfer_Otransfer__prod,axiom,
! [Alpha: $o > $o,F: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,F4: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,X4: produc5825016348098550007at_nat] :
( ! [A2: relational_fmla_a_b,B2: set_Pr1261947904930325089at_nat] : ( ord_less_eq_o @ ( Alpha @ ( F @ A2 @ B2 ) ) @ ( F4 @ A2 @ B2 ) )
=> ( ord_less_eq_o @ ( Alpha @ ( produc5861013030283631554_nat_o @ F @ X4 ) ) @ ( produc5861013030283631554_nat_o @ F4 @ X4 ) ) ) ).
% transfer.transfer_prod
thf(fact_831_transfer_Otransfer__prod,axiom,
! [Alpha: $o > $o,F: $o > $o > $o,F4: $o > $o > $o,X4: product_prod_o_o] :
( ! [A2: $o,B2: $o] : ( ord_less_eq_o @ ( Alpha @ ( F @ A2 @ B2 ) ) @ ( F4 @ A2 @ B2 ) )
=> ( ord_less_eq_o @ ( Alpha @ ( produc6197397395684419436_o_o_o @ F @ X4 ) ) @ ( produc6197397395684419436_o_o_o @ F4 @ X4 ) ) ) ).
% transfer.transfer_prod
thf(fact_832_ssubst__Pair__rhs,axiom,
! [R3: relational_fmla_a_b,S4: set_Pr1261947904930325089at_nat,R: set_Pr2645174627780777389at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ R3 @ S4 ) @ R )
=> ( ( S3 = S4 )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_833_ssubst__Pair__rhs,axiom,
! [R3: $o,S4: $o,R: set_Product_prod_o_o,S3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R3 @ S4 ) @ R )
=> ( ( S3 = S4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_834_prop__restrict,axiom,
! [X4: product_prod_o_o,Z5: set_Product_prod_o_o,X5: set_Product_prod_o_o,P2: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ X4 @ Z5 )
=> ( ( ord_le192195778054999463od_o_o @ Z5
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_835_prop__restrict,axiom,
! [X4: nat,Z5: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X4 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_836_prop__restrict,axiom,
! [X4: produc5825016348098550007at_nat,Z5: set_Pr2645174627780777389at_nat,X5: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( member6956540099943067662at_nat @ X4 @ Z5 )
=> ( ( ord_le8520675249591772685at_nat @ Z5
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_837_Collect__restrict,axiom,
! [X5: set_Product_prod_o_o,P2: product_prod_o_o > $o] :
( ord_le192195778054999463od_o_o
@ ( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_838_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_839_Collect__restrict,axiom,
! [X5: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o] :
( ord_le8520675249591772685at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_840_trancl__multi__insert2,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,M: $o,X5: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( transitive_trancl_o
@ ( sup_su5769328420594410459od_o_o @ R3
@ ( product_Sigma_o_o @ ( insert_o @ M @ bot_bot_set_o )
@ ^ [Uu: $o] : X5 ) ) ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ M ) @ ( transitive_rtrancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_841_trancl__multi__insert2,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,M: nat,X5: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B )
@ ( transi6264000038957366511cl_nat
@ ( sup_su6327502436637775413at_nat @ R3
@ ( produc457027306803732586at_nat @ ( insert_nat @ M @ bot_bot_set_nat )
@ ^ [Uu: nat] : X5 ) ) ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ M ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_842_trancl__multi__insert2,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,M: product_prod_o_o,X5: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B )
@ ( transi1494748803203666376od_o_o
@ ( sup_su4749226654166001307od_o_o @ R3
@ ( produc6514336722952383752od_o_o @ ( insert6201435330877294327od_o_o @ M @ bot_bo7073875226086086771od_o_o )
@ ^ [Uu: product_prod_o_o] : X5 ) ) ) )
=> ( ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( transi1494748803203666376od_o_o @ R3 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ X5 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ M ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X3 @ B ) @ ( transi4448435615503354666od_o_o @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_843_trancl__multi__insert2,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,R3: set_Pr3205448351804440111at_nat,M: produc5825016348098550007at_nat,X5: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B )
@ ( transi5629900152392516758at_nat
@ ( sup_su220504503009578499at_nat @ R3
@ ( produc984725824077784504at_nat @ ( insert3260075854425521959at_nat @ M @ bot_bo1973934853101755969at_nat )
@ ^ [Uu: produc5825016348098550007at_nat] : X5 ) ) ) )
=> ( ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ ( transi5629900152392516758at_nat @ R3 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ X5 )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ M ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X3 @ B ) @ ( transi7565848034366790580at_nat @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_844_trancl__multi__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,X5: set_o,M: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( transitive_trancl_o
@ ( sup_su5769328420594410459od_o_o @ R3
@ ( product_Sigma_o_o @ X5
@ ^ [Uu: $o] : ( insert_o @ M @ bot_bot_set_o ) ) ) ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_trancl_o @ R3 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ X3 ) @ ( transitive_rtrancl_o @ R3 ) )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ M @ B ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_845_trancl__multi__insert,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,X5: set_nat,M: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B )
@ ( transi6264000038957366511cl_nat
@ ( sup_su6327502436637775413at_nat @ R3
@ ( produc457027306803732586at_nat @ X5
@ ^ [Uu: nat] : ( insert_nat @ M @ bot_bot_set_nat ) ) ) ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_846_trancl__multi__insert,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,X5: set_Product_prod_o_o,M: product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B )
@ ( transi1494748803203666376od_o_o
@ ( sup_su4749226654166001307od_o_o @ R3
@ ( produc6514336722952383752od_o_o @ X5
@ ^ [Uu: product_prod_o_o] : ( insert6201435330877294327od_o_o @ M @ bot_bo7073875226086086771od_o_o ) ) ) ) )
=> ( ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( transi1494748803203666376od_o_o @ R3 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ X5 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ X3 ) @ ( transi4448435615503354666od_o_o @ R3 ) )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ M @ B ) @ ( transi4448435615503354666od_o_o @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_847_trancl__multi__insert,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,R3: set_Pr3205448351804440111at_nat,X5: set_Pr2645174627780777389at_nat,M: produc5825016348098550007at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B )
@ ( transi5629900152392516758at_nat
@ ( sup_su220504503009578499at_nat @ R3
@ ( produc984725824077784504at_nat @ X5
@ ^ [Uu: produc5825016348098550007at_nat] : ( insert3260075854425521959at_nat @ M @ bot_bo1973934853101755969at_nat ) ) ) ) )
=> ( ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ ( transi5629900152392516758at_nat @ R3 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ X5 )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ X3 ) @ ( transi7565848034366790580at_nat @ R3 ) )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ M @ B ) @ ( transi7565848034366790580at_nat @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_848_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A7: set_nat] : ( minus_minus_set_nat @ A7 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_849_remove__def,axiom,
( remove1047829871351731212od_o_o
= ( ^ [X: product_prod_o_o,A7: set_Product_prod_o_o] : ( minus_7052513330764642158od_o_o @ A7 @ ( insert6201435330877294327od_o_o @ X @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% remove_def
thf(fact_850_remove__def,axiom,
( remove6464231290375406418at_nat
= ( ^ [X: produc5825016348098550007at_nat,A7: set_Pr2645174627780777389at_nat] : ( minus_6698950876951835142at_nat @ A7 @ ( insert3260075854425521959at_nat @ X @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% remove_def
thf(fact_851_trancl__insert,axiom,
! [Y3: $o,X4: $o,R3: set_Product_prod_o_o] :
( ( transitive_trancl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ Y3 @ X4 ) @ R3 ) )
= ( sup_su5769328420594410459od_o_o @ ( transitive_trancl_o @ R3 )
@ ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [A3: $o,B3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ Y3 ) @ ( transitive_rtrancl_o @ R3 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ B3 ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_852_Un__iff,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( sup_su6099978769595272409at_nat @ A4 @ B5 ) )
= ( ( member6956540099943067662at_nat @ C @ A4 )
| ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_853_Un__iff,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ C @ A4 )
| ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_854_Un__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
| ( member_nat @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_855_UnCI,axiom,
! [C: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( ~ ( member6956540099943067662at_nat @ C @ B5 )
=> ( member6956540099943067662at_nat @ C @ A4 ) )
=> ( member6956540099943067662at_nat @ C @ ( sup_su6099978769595272409at_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_856_UnCI,axiom,
! [C: product_prod_o_o,B5: set_Product_prod_o_o,A4: set_Product_prod_o_o] :
( ( ~ ( member7466972457876170832od_o_o @ C @ B5 )
=> ( member7466972457876170832od_o_o @ C @ A4 ) )
=> ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_857_UnCI,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ A4 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_858_member__remove,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ X4 @ ( remove6464231290375406418at_nat @ Y3 @ A4 ) )
= ( ( member6956540099943067662at_nat @ X4 @ A4 )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_859_member__remove,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X4 @ ( remove1047829871351731212od_o_o @ Y3 @ A4 ) )
= ( ( member7466972457876170832od_o_o @ X4 @ A4 )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_860_member__remove,axiom,
! [X4: nat,Y3: nat,A4: set_nat] :
( ( member_nat @ X4 @ ( remove_nat @ Y3 @ A4 ) )
= ( ( member_nat @ X4 @ A4 )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_861_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_862_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ A @ bot_bo7073875226086086771od_o_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_863_sup__bot_Oright__neutral,axiom,
! [A: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A @ bot_bo1973934853101755969at_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_864_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_865_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( sup_su5769328420594410459od_o_o @ A @ B ) )
= ( ( A = bot_bo7073875226086086771od_o_o )
& ( B = bot_bo7073875226086086771od_o_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_866_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( sup_su6099978769595272409at_nat @ A @ B ) )
= ( ( A = bot_bo1973934853101755969at_nat )
& ( B = bot_bo1973934853101755969at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_867_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_868_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ bot_bo7073875226086086771od_o_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_869_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ bot_bo1973934853101755969at_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_870_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_871_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
( ( ( sup_su5769328420594410459od_o_o @ A @ B )
= bot_bo7073875226086086771od_o_o )
= ( ( A = bot_bo7073875226086086771od_o_o )
& ( B = bot_bo7073875226086086771od_o_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_872_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ( ( sup_su6099978769595272409at_nat @ A @ B )
= bot_bo1973934853101755969at_nat )
= ( ( A = bot_bo1973934853101755969at_nat )
& ( B = bot_bo1973934853101755969at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_873_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y3 )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_874_sup__eq__bot__iff,axiom,
! [X4: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
( ( ( sup_su5769328420594410459od_o_o @ X4 @ Y3 )
= bot_bo7073875226086086771od_o_o )
= ( ( X4 = bot_bo7073875226086086771od_o_o )
& ( Y3 = bot_bo7073875226086086771od_o_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_875_sup__eq__bot__iff,axiom,
! [X4: set_Pr2645174627780777389at_nat,Y3: set_Pr2645174627780777389at_nat] :
( ( ( sup_su6099978769595272409at_nat @ X4 @ Y3 )
= bot_bo1973934853101755969at_nat )
= ( ( X4 = bot_bo1973934853101755969at_nat )
& ( Y3 = bot_bo1973934853101755969at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_876_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_877_bot__eq__sup__iff,axiom,
! [X4: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( sup_su5769328420594410459od_o_o @ X4 @ Y3 ) )
= ( ( X4 = bot_bo7073875226086086771od_o_o )
& ( Y3 = bot_bo7073875226086086771od_o_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_878_bot__eq__sup__iff,axiom,
! [X4: set_Pr2645174627780777389at_nat,Y3: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( sup_su6099978769595272409at_nat @ X4 @ Y3 ) )
= ( ( X4 = bot_bo1973934853101755969at_nat )
& ( Y3 = bot_bo1973934853101755969at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_879_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_880_sup__bot__right,axiom,
! [X4: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ X4 @ bot_bo7073875226086086771od_o_o )
= X4 ) ).
% sup_bot_right
thf(fact_881_sup__bot__right,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ X4 @ bot_bo1973934853101755969at_nat )
= X4 ) ).
% sup_bot_right
thf(fact_882_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_883_sup__bot__left,axiom,
! [X4: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ bot_bo7073875226086086771od_o_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_884_sup__bot__left,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ bot_bo1973934853101755969at_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_885_Un__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ( A4 = bot_bot_set_nat )
& ( B5 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_886_Un__empty,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( sup_su5769328420594410459od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o )
= ( ( A4 = bot_bo7073875226086086771od_o_o )
& ( B5 = bot_bo7073875226086086771od_o_o ) ) ) ).
% Un_empty
thf(fact_887_Un__empty,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( sup_su6099978769595272409at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat )
= ( ( A4 = bot_bo1973934853101755969at_nat )
& ( B5 = bot_bo1973934853101755969at_nat ) ) ) ).
% Un_empty
thf(fact_888_finite__Un,axiom,
! [F4: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ F4 @ G2 ) )
= ( ( finite6790245451575510286at_nat @ F4 )
& ( finite6790245451575510286at_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_889_insert__Times__insert,axiom,
! [A: relational_fmla_a_b,A4: set_Re381260168593705685la_a_b,B: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
( ( produc7082850253950630328at_nat @ ( insert7010464514620295119la_a_b @ A @ A4 )
@ ^ [Uu: relational_fmla_a_b] : ( insert9200635055090092081at_nat @ B @ B5 ) )
= ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ A @ B )
@ ( sup_su6099978769595272409at_nat
@ ( produc7082850253950630328at_nat @ A4
@ ^ [Uu: relational_fmla_a_b] : ( insert9200635055090092081at_nat @ B @ B5 ) )
@ ( produc7082850253950630328at_nat @ ( insert7010464514620295119la_a_b @ A @ A4 )
@ ^ [Uu: relational_fmla_a_b] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_890_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: $o,B5: set_o] :
( ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( sup_su5769328420594410459od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
@ ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_891_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_892_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ X4 @ bot_bo7073875226086086771od_o_o )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_893_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ X4 @ bot_bo1973934853101755969at_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_894_Un__empty__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_895_Un__empty__left,axiom,
! [B5: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ bot_bo7073875226086086771od_o_o @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_896_Un__empty__left,axiom,
! [B5: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ bot_bo1973934853101755969at_nat @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_897_Un__empty__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Un_empty_right
thf(fact_898_Un__empty__right,axiom,
! [A4: set_Product_prod_o_o] :
( ( sup_su5769328420594410459od_o_o @ A4 @ bot_bo7073875226086086771od_o_o )
= A4 ) ).
% Un_empty_right
thf(fact_899_Un__empty__right,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( sup_su6099978769595272409at_nat @ A4 @ bot_bo1973934853101755969at_nat )
= A4 ) ).
% Un_empty_right
thf(fact_900_finite__UnI,axiom,
! [F4: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ F4 )
=> ( ( finite6790245451575510286at_nat @ G2 )
=> ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ F4 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_901_Un__infinite,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ~ ( finite6790245451575510286at_nat @ S )
=> ~ ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_902_infinite__Un,axiom,
! [S: set_Pr2645174627780777389at_nat,T: set_Pr2645174627780777389at_nat] :
( ( ~ ( finite6790245451575510286at_nat @ ( sup_su6099978769595272409at_nat @ S @ T ) ) )
= ( ~ ( finite6790245451575510286at_nat @ S )
| ~ ( finite6790245451575510286at_nat @ T ) ) ) ).
% infinite_Un
thf(fact_903_UnI2,axiom,
! [C: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ B5 )
=> ( member6956540099943067662at_nat @ C @ ( sup_su6099978769595272409at_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_904_UnI2,axiom,
! [C: product_prod_o_o,B5: set_Product_prod_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ B5 )
=> ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_905_UnI2,axiom,
! [C: nat,B5: set_nat,A4: set_nat] :
( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_906_UnI1,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ A4 )
=> ( member6956540099943067662at_nat @ C @ ( sup_su6099978769595272409at_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_907_UnI1,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ A4 )
=> ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_908_UnI1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_909_UnE,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( sup_su6099978769595272409at_nat @ A4 @ B5 ) )
=> ( ~ ( member6956540099943067662at_nat @ C @ A4 )
=> ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% UnE
thf(fact_910_UnE,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) )
=> ( ~ ( member7466972457876170832od_o_o @ C @ A4 )
=> ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% UnE
thf(fact_911_UnE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ~ ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% UnE
thf(fact_912_Un__def,axiom,
( sup_su5769328420594410459od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A7 )
| ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_913_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A7 )
| ( member_nat @ X @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_914_Un__def,axiom,
( sup_su6099978769595272409at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A7 )
| ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_915_Collect__disj__eq,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P2 @ X )
| ( Q @ X ) ) )
= ( sup_su6099978769595272409at_nat @ ( collec7697611494764367948at_nat @ P2 ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_916_insert__def,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat] :
( sup_su6099978769595272409at_nat
@ ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_917_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_918_insert__is__Un,axiom,
( insert6201435330877294327od_o_o
= ( ^ [A3: product_prod_o_o] : ( sup_su5769328420594410459od_o_o @ ( insert6201435330877294327od_o_o @ A3 @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% insert_is_Un
thf(fact_919_insert__is__Un,axiom,
( insert3260075854425521959at_nat
= ( ^ [A3: produc5825016348098550007at_nat] : ( sup_su6099978769595272409at_nat @ ( insert3260075854425521959at_nat @ A3 @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_920_Un__singleton__iff,axiom,
! [A4: set_nat,B5: set_nat,X4: nat] :
( ( ( sup_sup_set_nat @ A4 @ B5 )
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_921_Un__singleton__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( ( sup_su5769328420594410459od_o_o @ A4 @ B5 )
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= ( ( ( A4 = bot_bo7073875226086086771od_o_o )
& ( B5
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) )
| ( ( A4
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
& ( B5 = bot_bo7073875226086086771od_o_o ) )
| ( ( A4
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
& ( B5
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_922_Un__singleton__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( ( sup_su6099978769595272409at_nat @ A4 @ B5 )
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
= ( ( ( A4 = bot_bo1973934853101755969at_nat )
& ( B5
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) )
| ( ( A4
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
& ( B5 = bot_bo1973934853101755969at_nat ) )
| ( ( A4
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
& ( B5
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_923_singleton__Un__iff,axiom,
! [X4: nat,A4: set_nat,B5: set_nat] :
( ( ( insert_nat @ X4 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_924_singleton__Un__iff,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o )
= ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) )
= ( ( ( A4 = bot_bo7073875226086086771od_o_o )
& ( B5
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) )
| ( ( A4
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
& ( B5 = bot_bo7073875226086086771od_o_o ) )
| ( ( A4
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
& ( B5
= ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_925_singleton__Un__iff,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat )
= ( sup_su6099978769595272409at_nat @ A4 @ B5 ) )
= ( ( ( A4 = bot_bo1973934853101755969at_nat )
& ( B5
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) )
| ( ( A4
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
& ( B5 = bot_bo1973934853101755969at_nat ) )
| ( ( A4
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) )
& ( B5
= ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_926_rtrancl__Un__separator__converseE,axiom,
! [A: $o,B: $o,P2: set_Product_prod_o_o,Q: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ ( sup_su5769328420594410459od_o_o @ P2 @ Q ) ) )
=> ( ! [X3: $o,Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ ( transitive_rtrancl_o @ P2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X3 ) @ Q )
=> ( Y = X3 ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_927_rtrancl__Un__separatorE,axiom,
! [A: $o,B: $o,P2: set_Product_prod_o_o,Q: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ ( sup_su5769328420594410459od_o_o @ P2 @ Q ) ) )
=> ( ! [X3: $o,Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ X3 ) @ ( transitive_rtrancl_o @ P2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ Q )
=> ( X3 = Y ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( transitive_rtrancl_o @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_928_trancl__union__outside,axiom,
! [V: $o,W: $o,E: set_Product_prod_o_o,U3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( sup_su5769328420594410459od_o_o @ E @ U3 ) ) )
=> ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ E ) )
=> ? [X3: $o,Y: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ X3 ) @ ( transitive_rtrancl_o @ ( sup_su5769328420594410459od_o_o @ E @ U3 ) ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ U3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ W ) @ ( transitive_rtrancl_o @ ( sup_su5769328420594410459od_o_o @ E @ U3 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_929_rtrancl__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( transitive_rtrancl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( sup_su5769328420594410459od_o_o @ ( transitive_rtrancl_o @ R3 )
@ ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ A ) @ ( transitive_rtrancl_o @ R3 ) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ Y4 ) @ ( transitive_rtrancl_o @ R3 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_930_trancl__insert2,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( transitive_trancl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( sup_su5769328420594410459od_o_o @ ( transitive_trancl_o @ R3 )
@ ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [X: $o,Y4: $o] :
( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ A ) @ ( transitive_trancl_o @ R3 ) )
| ( X = A ) )
& ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ Y4 ) @ ( transitive_trancl_o @ R3 ) )
| ( Y4 = B ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_931_set__update,axiom,
! [K: $o,V: $o,Al: assoc_assoc_list_o_o] :
( ( assoc_set_o_o @ ( assoc_update_o_o @ K @ V @ Al ) )
= ( sup_su5769328420594410459od_o_o
@ ( minus_7052513330764642158od_o_o @ ( assoc_set_o_o @ Al )
@ ( product_Sigma_o_o @ ( insert_o @ K @ bot_bot_set_o )
@ ^ [Uu: $o] : top_top_set_o ) )
@ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ K @ V ) @ bot_bo7073875226086086771od_o_o ) ) ) ).
% set_update
thf(fact_932_set__update,axiom,
! [K: relational_fmla_a_b,V: set_Pr1261947904930325089at_nat,Al: assoc_222979904942530595at_nat] :
( ( assoc_4706750598316192965at_nat @ ( assoc_9021538453064944208at_nat @ K @ V @ Al ) )
= ( sup_su6099978769595272409at_nat
@ ( minus_6698950876951835142at_nat @ ( assoc_4706750598316192965at_nat @ Al )
@ ( produc7082850253950630328at_nat @ ( insert7010464514620295119la_a_b @ K @ bot_bo4495933725496725865la_a_b )
@ ^ [Uu: relational_fmla_a_b] : top_to7629004291339433233at_nat ) )
@ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ K @ V ) @ bot_bo1973934853101755969at_nat ) ) ) ).
% set_update
thf(fact_933_Field__insert,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) )
= ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R3 ) ) ) ).
% Field_insert
thf(fact_934_Field__insert,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R3 ) ) ) ).
% Field_insert
thf(fact_935_Field__insert,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o] :
( ( field_7351064555842910324od_o_o @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ R3 ) )
= ( sup_su5769328420594410459od_o_o @ ( insert6201435330877294327od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) ) @ ( field_7351064555842910324od_o_o @ R3 ) ) ) ).
% Field_insert
thf(fact_936_Field__insert,axiom,
! [A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat,R3: set_Pr3205448351804440111at_nat] :
( ( field_6081320236486121706at_nat @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ R3 ) )
= ( sup_su6099978769595272409at_nat @ ( insert3260075854425521959at_nat @ A @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) ) @ ( field_6081320236486121706at_nat @ R3 ) ) ) ).
% Field_insert
thf(fact_937_Sup__fin_Oremove,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X4 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_938_Sup__fin_Oremove,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( ( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ A4 )
= X4 ) )
& ( ( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
!= bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ A4 )
= ( sup_su8920606455873353467od_o_o @ X4 @ ( lattic8473092939728248030od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_939_Sup__fin_Oinsert__remove,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A4 ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A4 ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_940_Sup__fin_Oinsert__remove,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( ( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) )
= X4 ) )
& ( ( ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
!= bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) )
= ( sup_su8920606455873353467od_o_o @ X4 @ ( lattic8473092939728248030od_o_o @ ( minus_7052513330764642158od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_941_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_942_Field__empty,axiom,
( ( field_7351064555842910324od_o_o @ bot_bo2881837355931550515od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Field_empty
thf(fact_943_Field__empty,axiom,
( ( field_6081320236486121706at_nat @ bot_bo2147731989722242715at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Field_empty
thf(fact_944_Field__empty,axiom,
( ( field_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Field_empty
thf(fact_945_Sup__fin_Osingleton,axiom,
! [X4: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% Sup_fin.singleton
thf(fact_946_Sup__fin_Osingleton,axiom,
! [X4: product_prod_o_o] :
( ( lattic8473092939728248030od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) )
= X4 ) ).
% Sup_fin.singleton
thf(fact_947_finite__Field__eq__finite,axiom,
! [R: set_Pr3205448351804440111at_nat] :
( ( finite6790245451575510286at_nat @ ( field_6081320236486121706at_nat @ R ) )
= ( finite8185985059073859960at_nat @ R ) ) ).
% finite_Field_eq_finite
thf(fact_948_Sup__fin_Oinsert,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A4 ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_949_Sup__fin_Oinsert,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) )
= ( sup_su8920606455873353467od_o_o @ X4 @ ( lattic8473092939728248030od_o_o @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_950_FieldI2,axiom,
! [I2: produc5825016348098550007at_nat,J: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ I2 @ J ) @ R )
=> ( member6956540099943067662at_nat @ J @ ( field_6081320236486121706at_nat @ R ) ) ) ).
% FieldI2
thf(fact_951_FieldI2,axiom,
! [I2: product_prod_o_o,J: product_prod_o_o,R: set_Pr7011910401827753159od_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ I2 @ J ) @ R )
=> ( member7466972457876170832od_o_o @ J @ ( field_7351064555842910324od_o_o @ R ) ) ) ).
% FieldI2
thf(fact_952_FieldI2,axiom,
! [I2: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R )
=> ( member_nat @ J @ ( field_nat @ R ) ) ) ).
% FieldI2
thf(fact_953_FieldI2,axiom,
! [I2: $o,J: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R )
=> ( member_o @ J @ ( field_o @ R ) ) ) ).
% FieldI2
thf(fact_954_FieldI1,axiom,
! [I2: produc5825016348098550007at_nat,J: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ I2 @ J ) @ R )
=> ( member6956540099943067662at_nat @ I2 @ ( field_6081320236486121706at_nat @ R ) ) ) ).
% FieldI1
thf(fact_955_FieldI1,axiom,
! [I2: product_prod_o_o,J: product_prod_o_o,R: set_Pr7011910401827753159od_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ I2 @ J ) @ R )
=> ( member7466972457876170832od_o_o @ I2 @ ( field_7351064555842910324od_o_o @ R ) ) ) ).
% FieldI1
thf(fact_956_FieldI1,axiom,
! [I2: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R )
=> ( member_nat @ I2 @ ( field_nat @ R ) ) ) ).
% FieldI1
thf(fact_957_FieldI1,axiom,
! [I2: $o,J: $o,R: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R )
=> ( member_o @ I2 @ ( field_o @ R ) ) ) ).
% FieldI1
thf(fact_958_sup__set__def,axiom,
( sup_su5769328420594410459od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ( sup_su9111707261624351938_o_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A7 )
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_959_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A7 )
@ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_960_sup__set__def,axiom,
( sup_su6099978769595272409at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ( sup_su2875849708676129004_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ A7 )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_961_sup__Un__eq,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( sup_su2875849708676129004_nat_o
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ R )
@ ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ S ) )
= ( ^ [X: produc5825016348098550007at_nat] : ( member6956540099943067662at_nat @ X @ ( sup_su6099978769595272409at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_962_sup__Un__eq,axiom,
! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( sup_su9111707261624351938_o_o_o
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ R )
@ ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ S ) )
= ( ^ [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ ( sup_su5769328420594410459od_o_o @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_963_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_964_finite__Field,axiom,
! [R3: set_Pr3205448351804440111at_nat] :
( ( finite8185985059073859960at_nat @ R3 )
=> ( finite6790245451575510286at_nat @ ( field_6081320236486121706at_nat @ R3 ) ) ) ).
% finite_Field
thf(fact_965_Sup__fin_OcoboundedI,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ord_le2636248316962427079od_o_o @ A @ ( lattic8473092939728248030od_o_o @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_966_Sup__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_967_sup__Un__eq2,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr2645174627780777389at_nat] :
( ( sup_su7712494149945372102_nat_o
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ R )
@ ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ S ) )
= ( ^ [X: relational_fmla_a_b,Y4: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ ( sup_su6099978769595272409at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_968_sup__Un__eq2,axiom,
! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( sup_sup_o_o_o
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R )
@ ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ S ) )
= ( ^ [X: $o,Y4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ ( sup_su5769328420594410459od_o_o @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_969_Sup__fin_Oin__idem,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( sup_su8920606455873353467od_o_o @ X4 @ ( lattic8473092939728248030od_o_o @ A4 ) )
= ( lattic8473092939728248030od_o_o @ A4 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_970_Sup__fin_Oin__idem,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X4 @ A4 )
=> ( ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A4 ) )
= ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_971_Sup__fin_Obounded__iff,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X4 )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_972_Sup__fin_Obounded__iff,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( ord_le2636248316962427079od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ X4 )
= ( ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
=> ( ord_le2636248316962427079od_o_o @ X @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_973_Sup__fin_OboundedI,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ord_less_eq_nat @ A2 @ X4 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_974_Sup__fin_OboundedI,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ord_le2636248316962427079od_o_o @ A2 @ X4 ) )
=> ( ord_le2636248316962427079od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_975_Sup__fin_OboundedE,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X4 )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A4 )
=> ( ord_less_eq_nat @ A10 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_976_Sup__fin_OboundedE,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( ord_le2636248316962427079od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ X4 )
=> ! [A10: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A10 @ A4 )
=> ( ord_le2636248316962427079od_o_o @ A10 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_977_finite__Linear__order__induct,axiom,
! [R3: set_Pr3205448351804440111at_nat,X4: produc5825016348098550007at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( order_3226272362920281346at_nat @ ( field_6081320236486121706at_nat @ R3 ) @ R3 )
=> ( ( member6956540099943067662at_nat @ X4 @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( finite8185985059073859960at_nat @ R3 )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ! [Y5: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y5 @ ( order_2677216917948982476at_nat @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_978_finite__Linear__order__induct,axiom,
! [R3: set_Pr7011910401827753159od_o_o,X4: product_prod_o_o,P2: product_prod_o_o > $o] :
( ( order_7019990371155863132od_o_o @ ( field_7351064555842910324od_o_o @ R3 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ X4 @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( finite5612094862846137872od_o_o @ R3 )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ! [Y5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y5 @ ( order_2107371358090484754od_o_o @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_979_finite__Linear__order__induct,axiom,
! [R3: set_Pr1261947904930325089at_nat,X4: nat,P2: nat > $o] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R3 ) @ R3 )
=> ( ( member_nat @ X4 @ ( field_nat @ R3 ) )
=> ( ( finite6177210948735845034at_nat @ R3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( field_nat @ R3 ) )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( order_aboveS_nat @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_980_Sup__fin_Osubset__imp,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_981_Sup__fin_Osubset__imp,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( finite6120865539452801872od_o_o @ B5 )
=> ( ord_le2636248316962427079od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ ( lattic8473092939728248030od_o_o @ B5 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_982_Sup__fin_Osubset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B5 ) @ ( lattic1093996805478795353in_nat @ A4 ) )
= ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_983_Sup__fin_Osubset,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( B5 != bot_bo7073875226086086771od_o_o )
=> ( ( ord_le192195778054999463od_o_o @ B5 @ A4 )
=> ( ( sup_su8920606455873353467od_o_o @ ( lattic8473092939728248030od_o_o @ B5 ) @ ( lattic8473092939728248030od_o_o @ A4 ) )
= ( lattic8473092939728248030od_o_o @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_984_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_nat,X4: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X4 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A4 ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_985_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ~ ( member7466972457876170832od_o_o @ X4 @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ ( insert6201435330877294327od_o_o @ X4 @ A4 ) )
= ( sup_su8920606455873353467od_o_o @ X4 @ ( lattic8473092939728248030od_o_o @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_986_Sup__fin_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( insert_nat @ X3 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_987_Sup__fin_Oclosed,axiom,
! [A4: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ! [X3: product_prod_o_o,Y: product_prod_o_o] : ( member7466972457876170832od_o_o @ ( sup_su8920606455873353467od_o_o @ X3 @ Y ) @ ( insert6201435330877294327od_o_o @ X3 @ ( insert6201435330877294327od_o_o @ Y @ bot_bo7073875226086086771od_o_o ) ) )
=> ( member7466972457876170832od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_988_Sup__fin_Ounion,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B5 )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B5 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_989_Sup__fin_Ounion,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( A4 != bot_bo7073875226086086771od_o_o )
=> ( ( finite6120865539452801872od_o_o @ B5 )
=> ( ( B5 != bot_bo7073875226086086771od_o_o )
=> ( ( lattic8473092939728248030od_o_o @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) )
= ( sup_su8920606455873353467od_o_o @ ( lattic8473092939728248030od_o_o @ A4 ) @ ( lattic8473092939728248030od_o_o @ B5 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_990_Assoc__List_Oset__empty,axiom,
( ( assoc_set_o_o @ assoc_empty_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Assoc_List.set_empty
thf(fact_991_Assoc__List_Oset__empty,axiom,
( ( assoc_4706750598316192965at_nat @ assoc_5512385647536227482at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Assoc_List.set_empty
thf(fact_992_cofinal__def,axiom,
( bNF_Ca4363140458462388713inal_o
= ( ^ [A7: set_o,R2: set_Product_prod_o_o] :
! [X: $o] :
( ( member_o @ X @ ( field_o @ R2 ) )
=> ? [Y4: $o] :
( ( member_o @ Y4 @ A7 )
& ( X = (~ Y4) )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% cofinal_def
thf(fact_993_transymcl__insert__both__new,axiom,
! [A: produc5825016348098550007at_nat,Qeq2: set_Pr3205448351804440111at_nat,B: produc5825016348098550007at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ ( field_6081320236486121706at_nat @ Qeq2 ) )
=> ( ~ ( member6956540099943067662at_nat @ B @ ( field_6081320236486121706at_nat @ Qeq2 ) )
=> ( ( transy5404578198922738261at_nat @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ Qeq2 ) )
= ( sup_su220504503009578499at_nat @ ( transy5404578198922738261at_nat @ Qeq2 ) @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ A @ A ) @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ B @ B ) @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ A @ B ) @ ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ B @ A ) @ bot_bo2147731989722242715at_nat ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_994_transymcl__insert__both__new,axiom,
! [A: product_prod_o_o,Qeq2: set_Pr7011910401827753159od_o_o,B: product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ A @ ( field_7351064555842910324od_o_o @ Qeq2 ) )
=> ( ~ ( member7466972457876170832od_o_o @ B @ ( field_7351064555842910324od_o_o @ Qeq2 ) )
=> ( ( transy2585538050873104457od_o_o @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ Qeq2 ) )
= ( sup_su4749226654166001307od_o_o @ ( transy2585538050873104457od_o_o @ Qeq2 ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ A @ A ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ B @ B ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ ( insert3377906364125778615od_o_o @ ( produc3331439552374987607od_o_o @ B @ A ) @ bot_bo2881837355931550515od_o_o ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_995_transymcl__insert__both__new,axiom,
! [A: nat,Qeq2: set_Pr1261947904930325089at_nat,B: nat] :
( ~ ( member_nat @ A @ ( field_nat @ Qeq2 ) )
=> ( ~ ( member_nat @ B @ ( field_nat @ Qeq2 ) )
=> ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ Qeq2 ) )
= ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B @ B ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B @ A ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_996_transymcl__insert__both__new,axiom,
! [A: $o,Qeq2: set_Product_prod_o_o,B: $o] :
( ~ ( member_o @ A @ ( field_o @ Qeq2 ) )
=> ( ~ ( member_o @ B @ ( field_o @ Qeq2 ) )
=> ( ( transymcl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ Qeq2 ) )
= ( sup_su5769328420594410459od_o_o @ ( transymcl_o @ Qeq2 ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ B ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ A ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_997_transymcl__into_I2_J,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X4 ) @ ( transymcl_o @ R3 ) ) ) ).
% transymcl_into(2)
thf(fact_998_transymcl__into_I1_J,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transymcl_o @ R3 ) ) ) ).
% transymcl_into(1)
thf(fact_999_transymcl__self_I2_J,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Y3 ) @ ( transymcl_o @ R3 ) ) ) ).
% transymcl_self(2)
thf(fact_1000_transymcl__self_I1_J,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ X4 ) @ ( transymcl_o @ R3 ) ) ) ).
% transymcl_self(1)
thf(fact_1001_transymcl__sym,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transymcl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X4 ) @ ( transymcl_o @ R3 ) ) ) ).
% transymcl_sym
thf(fact_1002_transymcl__trans,axiom,
! [X4: $o,Y3: $o,R3: set_Product_prod_o_o,Z: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transymcl_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Z ) @ ( transymcl_o @ R3 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Z ) @ ( transymcl_o @ R3 ) ) ) ) ).
% transymcl_trans
thf(fact_1003_Field__transymcl__self,axiom,
! [A: produc5825016348098550007at_nat,Qeq2: set_Pr3205448351804440111at_nat] :
( ( member6956540099943067662at_nat @ A @ ( field_6081320236486121706at_nat @ Qeq2 ) )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ A ) @ ( transy5404578198922738261at_nat @ Qeq2 ) ) ) ).
% Field_transymcl_self
thf(fact_1004_Field__transymcl__self,axiom,
! [A: product_prod_o_o,Qeq2: set_Pr7011910401827753159od_o_o] :
( ( member7466972457876170832od_o_o @ A @ ( field_7351064555842910324od_o_o @ Qeq2 ) )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ A ) @ ( transy2585538050873104457od_o_o @ Qeq2 ) ) ) ).
% Field_transymcl_self
thf(fact_1005_Field__transymcl__self,axiom,
! [A: nat,Qeq2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( field_nat @ Qeq2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transymcl_nat @ Qeq2 ) ) ) ).
% Field_transymcl_self
thf(fact_1006_Field__transymcl__self,axiom,
! [A: $o,Qeq2: set_Product_prod_o_o] :
( ( member_o @ A @ ( field_o @ Qeq2 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( transymcl_o @ Qeq2 ) ) ) ).
% Field_transymcl_self
thf(fact_1007_transymcl__insert__same__class,axiom,
! [X4: $o,Y3: $o,Qeq2: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transymcl_o @ Qeq2 ) )
=> ( ( transymcl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ Qeq2 ) )
= ( transymcl_o @ Qeq2 ) ) ) ).
% transymcl_insert_same_class
thf(fact_1008_transymcl__insert,axiom,
! [A: $o,B: $o,Qeq2: set_Product_prod_o_o] :
( ( transymcl_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ Qeq2 ) )
= ( sup_su5769328420594410459od_o_o @ ( sup_su5769328420594410459od_o_o @ ( transymcl_o @ Qeq2 ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ B ) @ bot_bo7073875226086086771od_o_o ) ) ) @ ( minus_7052513330764642158od_o_o @ ( relcomp_o_o_o @ ( sup_su5769328420594410459od_o_o @ ( transymcl_o @ Qeq2 ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ B ) @ bot_bo7073875226086086771od_o_o ) ) ) @ ( relcomp_o_o_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ A ) @ bot_bo7073875226086086771od_o_o ) ) @ ( sup_su5769328420594410459od_o_o @ ( transymcl_o @ Qeq2 ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ B @ B ) @ bot_bo7073875226086086771od_o_o ) ) ) ) ) @ ( transymcl_o @ Qeq2 ) ) ) ) ).
% transymcl_insert
thf(fact_1009_subset__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_o_o,A: $o,B: $o] :
( ( order_preorder_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( member_o @ A @ ( field_o @ R3 ) )
=> ( ( member_o @ B @ ( field_o @ R3 ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R3 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1010_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4861654808422542329on_nat @ ( field_nat @ R3 ) @ R3 )
=> ( ( member_nat @ A @ ( field_nat @ R3 ) )
=> ( ( member_nat @ B @ ( field_nat @ R3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1011_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr7011910401827753159od_o_o,A: product_prod_o_o,B: product_prod_o_o] :
( ( order_3385673970029650238od_o_o @ ( field_7351064555842910324od_o_o @ R3 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ B @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) @ ( image_539601260427297683od_o_o @ R3 @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) ) )
= ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1012_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr3205448351804440111at_nat,A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat] :
( ( order_6739036243672267424at_nat @ ( field_6081320236486121706at_nat @ R3 ) @ R3 )
=> ( ( member6956540099943067662at_nat @ A @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( member6956540099943067662at_nat @ B @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ R3 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) @ ( image_4491765980459089603at_nat @ R3 @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) ) )
= ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ B @ A ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1013_finite__Partial__order__induct,axiom,
! [R3: set_Pr3205448351804440111at_nat,X4: produc5825016348098550007at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( order_7623158883673130818at_nat @ ( field_6081320236486121706at_nat @ R3 ) @ R3 )
=> ( ( member6956540099943067662at_nat @ X4 @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( finite8185985059073859960at_nat @ R3 )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ! [Y5: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y5 @ ( order_2677216917948982476at_nat @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1014_finite__Partial__order__induct,axiom,
! [R3: set_Pr7011910401827753159od_o_o,X4: product_prod_o_o,P2: product_prod_o_o > $o] :
( ( order_4683604368562820124od_o_o @ ( field_7351064555842910324od_o_o @ R3 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ X4 @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( finite5612094862846137872od_o_o @ R3 )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ! [Y5: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y5 @ ( order_2107371358090484754od_o_o @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1015_finite__Partial__order__induct,axiom,
! [R3: set_Pr1261947904930325089at_nat,X4: nat,P2: nat > $o] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R3 ) @ R3 )
=> ( ( member_nat @ X4 @ ( field_nat @ R3 ) )
=> ( ( finite6177210948735845034at_nat @ R3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( field_nat @ R3 ) )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( order_aboveS_nat @ R3 @ X3 ) )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1016_ImageI,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( ( member_nat @ A @ A4 )
=> ( member_nat @ B @ ( image_nat_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1017_ImageI,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,A4: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member_o @ A @ A4 )
=> ( member_o @ B @ ( image_o_o @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1018_ImageI,axiom,
! [A: product_prod_o_o,B: nat,R3: set_Pr6872669703010578012_o_nat,A4: set_Product_prod_o_o] :
( ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( member_nat @ B @ ( image_3423961300553328228_o_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1019_ImageI,axiom,
! [A: nat,B: product_prod_o_o,R3: set_Pr4076497759046268930od_o_o,A4: set_nat] :
( ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ R3 )
=> ( ( member_nat @ A @ A4 )
=> ( member7466972457876170832od_o_o @ B @ ( image_8931571761295819714od_o_o @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1020_ImageI,axiom,
! [A: product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A4: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ B @ ( image_539601260427297683od_o_o @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1021_ImageI,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,A4: set_Re381260168593705685la_a_b] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ R3 )
=> ( ( member4680049679412964150la_a_b @ A @ A4 )
=> ( member2643936169264416010at_nat @ B @ ( image_2866438327291862765at_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1022_ImageI,axiom,
! [A: produc5825016348098550007at_nat,B: nat,R3: set_Pr8118092986756011272at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ R3 )
=> ( ( member6956540099943067662at_nat @ A @ A4 )
=> ( member_nat @ B @ ( image_4683373685380921244at_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1023_ImageI,axiom,
! [A: nat,B: produc5825016348098550007at_nat,R3: set_Pr2590682000163020040at_nat,A4: set_nat] :
( ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ R3 )
=> ( ( member_nat @ A @ A4 )
=> ( member6956540099943067662at_nat @ B @ ( image_6961426772032973084at_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1024_ImageI,axiom,
! [A: produc5825016348098550007at_nat,B: product_prod_o_o,R3: set_Pr4444513774987008283od_o_o,A4: set_Pr2645174627780777389at_nat] :
( ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ R3 )
=> ( ( member6956540099943067662at_nat @ A @ A4 )
=> ( member7466972457876170832od_o_o @ B @ ( image_4792952963437465691od_o_o @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1025_ImageI,axiom,
! [A: product_prod_o_o,B: produc5825016348098550007at_nat,R3: set_Pr2403587099507754371at_nat,A4: set_Product_prod_o_o] :
( ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( member6956540099943067662at_nat @ B @ ( image_5920465547893864843at_nat @ R3 @ A4 ) ) ) ) ).
% ImageI
thf(fact_1026_Image__empty2,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( image_nat_nat @ R @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_1027_Image__empty2,axiom,
! [R: set_Pr4076497759046268930od_o_o] :
( ( image_8931571761295819714od_o_o @ R @ bot_bot_set_nat )
= bot_bo7073875226086086771od_o_o ) ).
% Image_empty2
thf(fact_1028_Image__empty2,axiom,
! [R: set_Pr2590682000163020040at_nat] :
( ( image_6961426772032973084at_nat @ R @ bot_bot_set_nat )
= bot_bo1973934853101755969at_nat ) ).
% Image_empty2
thf(fact_1029_Image__empty2,axiom,
! [R: set_Pr6872669703010578012_o_nat] :
( ( image_3423961300553328228_o_nat @ R @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_1030_Image__empty2,axiom,
! [R: set_Pr7011910401827753159od_o_o] :
( ( image_539601260427297683od_o_o @ R @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Image_empty2
thf(fact_1031_Image__empty2,axiom,
! [R: set_Pr2403587099507754371at_nat] :
( ( image_5920465547893864843at_nat @ R @ bot_bo7073875226086086771od_o_o )
= bot_bo1973934853101755969at_nat ) ).
% Image_empty2
thf(fact_1032_Image__empty2,axiom,
! [R: set_Pr8118092986756011272at_nat] :
( ( image_4683373685380921244at_nat @ R @ bot_bo1973934853101755969at_nat )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_1033_Image__empty2,axiom,
! [R: set_Pr4444513774987008283od_o_o] :
( ( image_4792952963437465691od_o_o @ R @ bot_bo1973934853101755969at_nat )
= bot_bo7073875226086086771od_o_o ) ).
% Image_empty2
thf(fact_1034_Image__empty2,axiom,
! [R: set_Pr3205448351804440111at_nat] :
( ( image_4491765980459089603at_nat @ R @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Image_empty2
thf(fact_1035_relcomp__empty1,axiom,
! [R: set_Product_prod_o_o] :
( ( relcomp_o_o_o @ bot_bo7073875226086086771od_o_o @ R )
= bot_bo7073875226086086771od_o_o ) ).
% relcomp_empty1
thf(fact_1036_relcomp__empty1,axiom,
! [R: set_Pr4329608150637261639at_nat] :
( ( relcom1463679224352411744at_nat @ bot_bo1973934853101755969at_nat @ R )
= bot_bo1973934853101755969at_nat ) ).
% relcomp_empty1
thf(fact_1037_relcomp__empty2,axiom,
! [R: set_Product_prod_o_o] :
( ( relcomp_o_o_o @ R @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% relcomp_empty2
thf(fact_1038_relcomp__empty2,axiom,
! [R: set_Pr389361089297601407la_a_b] :
( ( relcom1330972924984534752at_nat @ R @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% relcomp_empty2
thf(fact_1039_finite__relcomp,axiom,
! [R: set_Pr389361089297601407la_a_b,S: set_Pr2645174627780777389at_nat] :
( ( finite3172801315471850184la_a_b @ R )
=> ( ( finite6790245451575510286at_nat @ S )
=> ( finite6790245451575510286at_nat @ ( relcom1330972924984534752at_nat @ R @ S ) ) ) ) ).
% finite_relcomp
thf(fact_1040_finite__relcomp,axiom,
! [R: set_Pr2645174627780777389at_nat,S: set_Pr4329608150637261639at_nat] :
( ( finite6790245451575510286at_nat @ R )
=> ( ( finite4343798906461161616at_nat @ S )
=> ( finite6790245451575510286at_nat @ ( relcom1463679224352411744at_nat @ R @ S ) ) ) ) ).
% finite_relcomp
thf(fact_1041_Image__empty1,axiom,
! [X5: set_o] :
( ( image_o_o @ bot_bo7073875226086086771od_o_o @ X5 )
= bot_bot_set_o ) ).
% Image_empty1
thf(fact_1042_Image__empty1,axiom,
! [X5: set_Re381260168593705685la_a_b] :
( ( image_2866438327291862765at_nat @ bot_bo1973934853101755969at_nat @ X5 )
= bot_bo3083307316010499117at_nat ) ).
% Image_empty1
thf(fact_1043_Image__Collect__case__prod,axiom,
! [P2: relational_fmla_a_b > set_Pr1261947904930325089at_nat > $o,A4: set_Re381260168593705685la_a_b] :
( ( image_2866438327291862765at_nat @ ( collec7697611494764367948at_nat @ ( produc5861013030283631554_nat_o @ P2 ) ) @ A4 )
= ( collec5514110066124741708at_nat
@ ^ [Y4: set_Pr1261947904930325089at_nat] :
? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
& ( P2 @ X @ Y4 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_1044_Image__Collect__case__prod,axiom,
! [P2: $o > $o > $o,A4: set_o] :
( ( image_o_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ P2 ) ) @ A4 )
= ( collect_o
@ ^ [Y4: $o] :
? [X: $o] :
( ( member_o @ X @ A4 )
& ( P2 @ X @ Y4 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_1045_Image__singleton__iff,axiom,
! [B: $o,R3: set_Product_prod_o_o,A: $o] :
( ( member_o @ B @ ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1046_Image__singleton__iff,axiom,
! [B: nat,R3: set_Pr1261947904930325089at_nat,A: nat] :
( ( member_nat @ B @ ( image_nat_nat @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1047_Image__singleton__iff,axiom,
! [B: product_prod_o_o,R3: set_Pr4076497759046268930od_o_o,A: nat] :
( ( member7466972457876170832od_o_o @ B @ ( image_8931571761295819714od_o_o @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1048_Image__singleton__iff,axiom,
! [B: nat,R3: set_Pr6872669703010578012_o_nat,A: product_prod_o_o] :
( ( member_nat @ B @ ( image_3423961300553328228_o_nat @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1049_Image__singleton__iff,axiom,
! [B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( image_539601260427297683od_o_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1050_Image__singleton__iff,axiom,
! [B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,A: relational_fmla_a_b] :
( ( member2643936169264416010at_nat @ B @ ( image_2866438327291862765at_nat @ R3 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
= ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1051_Image__singleton__iff,axiom,
! [B: produc5825016348098550007at_nat,R3: set_Pr2590682000163020040at_nat,A: nat] :
( ( member6956540099943067662at_nat @ B @ ( image_6961426772032973084at_nat @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1052_Image__singleton__iff,axiom,
! [B: nat,R3: set_Pr8118092986756011272at_nat,A: produc5825016348098550007at_nat] :
( ( member_nat @ B @ ( image_4683373685380921244at_nat @ R3 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1053_Image__singleton__iff,axiom,
! [B: produc5825016348098550007at_nat,R3: set_Pr2403587099507754371at_nat,A: product_prod_o_o] :
( ( member6956540099943067662at_nat @ B @ ( image_5920465547893864843at_nat @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
= ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1054_Image__singleton__iff,axiom,
! [B: product_prod_o_o,R3: set_Pr4444513774987008283od_o_o,A: produc5825016348098550007at_nat] :
( ( member7466972457876170832od_o_o @ B @ ( image_4792952963437465691od_o_o @ R3 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) ) )
= ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1055_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R3 ) @ R3 )
=> ( ( member_nat @ A @ ( field_nat @ R3 ) )
=> ( ( member_nat @ B @ ( field_nat @ R3 ) )
=> ( ( ( image_nat_nat @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1056_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr7011910401827753159od_o_o,A: product_prod_o_o,B: product_prod_o_o] :
( ( order_4683604368562820124od_o_o @ ( field_7351064555842910324od_o_o @ R3 ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ A @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ B @ ( field_7351064555842910324od_o_o @ R3 ) )
=> ( ( ( image_539601260427297683od_o_o @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( image_539601260427297683od_o_o @ R3 @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1057_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr3205448351804440111at_nat,A: produc5825016348098550007at_nat,B: produc5825016348098550007at_nat] :
( ( order_7623158883673130818at_nat @ ( field_6081320236486121706at_nat @ R3 ) @ R3 )
=> ( ( member6956540099943067662at_nat @ A @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( member6956540099943067662at_nat @ B @ ( field_6081320236486121706at_nat @ R3 ) )
=> ( ( ( image_4491765980459089603at_nat @ R3 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( image_4491765980459089603at_nat @ R3 @ ( insert3260075854425521959at_nat @ B @ bot_bo1973934853101755969at_nat ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1058_rev__ImageI,axiom,
! [A: nat,A4: set_nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member_nat @ B @ ( image_nat_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1059_rev__ImageI,axiom,
! [A: $o,A4: set_o,B: $o,R3: set_Product_prod_o_o] :
( ( member_o @ A @ A4 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( member_o @ B @ ( image_o_o @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1060_rev__ImageI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: nat,R3: set_Pr6872669703010578012_o_nat] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ A @ B ) @ R3 )
=> ( member_nat @ B @ ( image_3423961300553328228_o_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1061_rev__ImageI,axiom,
! [A: nat,A4: set_nat,B: product_prod_o_o,R3: set_Pr4076497759046268930od_o_o] :
( ( member_nat @ A @ A4 )
=> ( ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_8931571761295819714od_o_o @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1062_rev__ImageI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_539601260427297683od_o_o @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1063_rev__ImageI,axiom,
! [A: relational_fmla_a_b,A4: set_Re381260168593705685la_a_b,B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat] :
( ( member4680049679412964150la_a_b @ A @ A4 )
=> ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ R3 )
=> ( member2643936169264416010at_nat @ B @ ( image_2866438327291862765at_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1064_rev__ImageI,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: nat,R3: set_Pr8118092986756011272at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ A @ B ) @ R3 )
=> ( member_nat @ B @ ( image_4683373685380921244at_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1065_rev__ImageI,axiom,
! [A: nat,A4: set_nat,B: produc5825016348098550007at_nat,R3: set_Pr2590682000163020040at_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B ) @ R3 )
=> ( member6956540099943067662at_nat @ B @ ( image_6961426772032973084at_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1066_rev__ImageI,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B: product_prod_o_o,R3: set_Pr4444513774987008283od_o_o] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ A @ B ) @ R3 )
=> ( member7466972457876170832od_o_o @ B @ ( image_4792952963437465691od_o_o @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1067_rev__ImageI,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B: produc5825016348098550007at_nat,R3: set_Pr2403587099507754371at_nat] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B ) @ R3 )
=> ( member6956540099943067662at_nat @ B @ ( image_5920465547893864843at_nat @ R3 @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_1068_Image__iff,axiom,
! [B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,A4: set_Re381260168593705685la_a_b] :
( ( member2643936169264416010at_nat @ B @ ( image_2866438327291862765at_nat @ R3 @ A4 ) )
= ( ? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A4 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1069_Image__iff,axiom,
! [B: $o,R3: set_Product_prod_o_o,A4: set_o] :
( ( member_o @ B @ ( image_o_o @ R3 @ A4 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A4 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ B ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1070_ImageE,axiom,
! [B: nat,R3: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R3 @ A4 ) )
=> ~ ! [X3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R3 )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1071_ImageE,axiom,
! [B: $o,R3: set_Product_prod_o_o,A4: set_o] :
( ( member_o @ B @ ( image_o_o @ R3 @ A4 ) )
=> ~ ! [X3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R3 )
=> ~ ( member_o @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1072_ImageE,axiom,
! [B: product_prod_o_o,R3: set_Pr4076497759046268930od_o_o,A4: set_nat] :
( ( member7466972457876170832od_o_o @ B @ ( image_8931571761295819714od_o_o @ R3 @ A4 ) )
=> ~ ! [X3: nat] :
( ( member4050856805493128931od_o_o @ ( produc1927143891212739326od_o_o @ X3 @ B ) @ R3 )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1073_ImageE,axiom,
! [B: nat,R3: set_Pr6872669703010578012_o_nat,A4: set_Product_prod_o_o] :
( ( member_nat @ B @ ( image_3423961300553328228_o_nat @ R3 @ A4 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member5772852833339327037_o_nat @ ( produc5642905467325023648_o_nat @ X3 @ B ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1074_ImageE,axiom,
! [B: product_prod_o_o,R3: set_Pr7011910401827753159od_o_o,A4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ B @ ( image_539601260427297683od_o_o @ R3 @ A4 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X3 @ B ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1075_ImageE,axiom,
! [B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,A4: set_Re381260168593705685la_a_b] :
( ( member2643936169264416010at_nat @ B @ ( image_2866438327291862765at_nat @ R3 @ A4 ) )
=> ~ ! [X3: relational_fmla_a_b] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X3 @ B ) @ R3 )
=> ~ ( member4680049679412964150la_a_b @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1076_ImageE,axiom,
! [B: produc5825016348098550007at_nat,R3: set_Pr2590682000163020040at_nat,A4: set_nat] :
( ( member6956540099943067662at_nat @ B @ ( image_6961426772032973084at_nat @ R3 @ A4 ) )
=> ~ ! [X3: nat] :
( ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ X3 @ B ) @ R3 )
=> ~ ( member_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1077_ImageE,axiom,
! [B: nat,R3: set_Pr8118092986756011272at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( member_nat @ B @ ( image_4683373685380921244at_nat @ R3 @ A4 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member6199062629724834769at_nat @ ( produc925257419237160800at_nat @ X3 @ B ) @ R3 )
=> ~ ( member6956540099943067662at_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1078_ImageE,axiom,
! [B: produc5825016348098550007at_nat,R3: set_Pr2403587099507754371at_nat,A4: set_Product_prod_o_o] :
( ( member6956540099943067662at_nat @ B @ ( image_5920465547893864843at_nat @ R3 @ A4 ) )
=> ~ ! [X3: product_prod_o_o] :
( ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ X3 @ B ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1079_ImageE,axiom,
! [B: product_prod_o_o,R3: set_Pr4444513774987008283od_o_o,A4: set_Pr2645174627780777389at_nat] :
( ( member7466972457876170832od_o_o @ B @ ( image_4792952963437465691od_o_o @ R3 @ A4 ) )
=> ~ ! [X3: produc5825016348098550007at_nat] :
( ( member5724791357406019196od_o_o @ ( produc7046591135137782167od_o_o @ X3 @ B ) @ R3 )
=> ~ ( member6956540099943067662at_nat @ X3 @ A4 ) ) ) ).
% ImageE
thf(fact_1080_relcompEpair,axiom,
! [A: relational_fmla_a_b,C: set_Pr1261947904930325089at_nat,R3: set_Pr389361089297601407la_a_b,S4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ C ) @ ( relcom1330972924984534752at_nat @ R3 @ S4 ) )
=> ~ ! [B2: relational_fmla_a_b] :
( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A @ B2 ) @ R3 )
=> ~ ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B2 @ C ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_1081_relcompEpair,axiom,
! [A: relational_fmla_a_b,C: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ C ) @ ( relcom1463679224352411744at_nat @ R3 @ S4 ) )
=> ~ ! [B2: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B2 ) @ R3 )
=> ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ C ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_1082_relcompEpair,axiom,
! [A: $o,C: $o,R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( relcomp_o_o_o @ R3 @ S4 ) )
=> ~ ! [B2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B2 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ C ) @ S4 ) ) ) ).
% relcompEpair
thf(fact_1083_relcompE,axiom,
! [Xz: produc5825016348098550007at_nat,R3: set_Pr389361089297601407la_a_b,S4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ Xz @ ( relcom1330972924984534752at_nat @ R3 @ S4 ) )
=> ~ ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b,Z2: set_Pr1261947904930325089at_nat] :
( ( Xz
= ( produc760948067230524457at_nat @ X3 @ Z2 ) )
=> ( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ X3 @ Y ) @ R3 )
=> ~ ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ Y @ Z2 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_1084_relcompE,axiom,
! [Xz: produc5825016348098550007at_nat,R3: set_Pr2645174627780777389at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member6956540099943067662at_nat @ Xz @ ( relcom1463679224352411744at_nat @ R3 @ S4 ) )
=> ~ ! [X3: relational_fmla_a_b,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( Xz
= ( produc760948067230524457at_nat @ X3 @ Z2 ) )
=> ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X3 @ Y ) @ R3 )
=> ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_1085_relcompE,axiom,
! [Xz: product_prod_o_o,R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Xz @ ( relcomp_o_o_o @ R3 @ S4 ) )
=> ~ ! [X3: $o,Y: $o,Z2: $o] :
( ( Xz
= ( product_Pair_o_o @ X3 @ Z2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ Z2 ) @ S4 ) ) ) ) ).
% relcompE
thf(fact_1086_relcomp_OrelcompI,axiom,
! [A: relational_fmla_a_b,B: relational_fmla_a_b,R3: set_Pr389361089297601407la_a_b,C: set_Pr1261947904930325089at_nat,S4: set_Pr2645174627780777389at_nat] :
( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A @ B ) @ R3 )
=> ( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B @ C ) @ S4 )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ C ) @ ( relcom1330972924984534752at_nat @ R3 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1087_relcomp_OrelcompI,axiom,
! [A: relational_fmla_a_b,B: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,C: set_Pr1261947904930325089at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B ) @ R3 )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ S4 )
=> ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ C ) @ ( relcom1463679224352411744at_nat @ R3 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1088_relcomp_OrelcompI,axiom,
! [A: $o,B: $o,R3: set_Product_prod_o_o,C: $o,S4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ C ) @ S4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ C ) @ ( relcomp_o_o_o @ R3 @ S4 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1089_relcomp_Osimps,axiom,
! [A1: relational_fmla_a_b,A22: set_Pr1261947904930325089at_nat,R3: set_Pr389361089297601407la_a_b,S4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A1 @ A22 ) @ ( relcom1330972924984534752at_nat @ R3 @ S4 ) )
= ( ? [A3: relational_fmla_a_b,B3: relational_fmla_a_b,C3: set_Pr1261947904930325089at_nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A3 @ B3 ) @ R3 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B3 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_1090_relcomp_Osimps,axiom,
! [A1: relational_fmla_a_b,A22: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A1 @ A22 ) @ ( relcom1463679224352411744at_nat @ R3 @ S4 ) )
= ( ? [A3: relational_fmla_a_b,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A3 @ B3 ) @ R3 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_1091_relcomp_Osimps,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( relcomp_o_o_o @ R3 @ S4 ) )
= ( ? [A3: $o,B3: $o,C3: $o] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ C3 ) @ S4 ) ) ) ) ).
% relcomp.simps
thf(fact_1092_relcomp_Ocases,axiom,
! [A1: relational_fmla_a_b,A22: set_Pr1261947904930325089at_nat,R3: set_Pr389361089297601407la_a_b,S4: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A1 @ A22 ) @ ( relcom1330972924984534752at_nat @ R3 @ S4 ) )
=> ~ ! [B2: relational_fmla_a_b] :
( ( member3635249065003913160la_a_b @ ( produc108381030462314967la_a_b @ A1 @ B2 ) @ R3 )
=> ~ ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ B2 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_1093_relcomp_Ocases,axiom,
! [A1: relational_fmla_a_b,A22: set_Pr1261947904930325089at_nat,R3: set_Pr2645174627780777389at_nat,S4: set_Pr4329608150637261639at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A1 @ A22 ) @ ( relcom1463679224352411744at_nat @ R3 @ S4 ) )
=> ~ ! [B2: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A1 @ B2 ) @ R3 )
=> ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_1094_relcomp_Ocases,axiom,
! [A1: $o,A22: $o,R3: set_Product_prod_o_o,S4: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ A22 ) @ ( relcomp_o_o_o @ R3 @ S4 ) )
=> ~ ! [B2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B2 ) @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ A22 ) @ S4 ) ) ) ).
% relcomp.cases
thf(fact_1095_max__ext__compat,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R @ S ) @ R )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R ) ) ) ).
% max_ext_compat
thf(fact_1096_max__ext__compat,axiom,
! [R: set_Pr7011910401827753159od_o_o,S: set_Pr7011910401827753159od_o_o] :
( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R @ S ) @ R )
=> ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( max_ex7572203241849361634od_o_o @ R ) @ ( sup_su4078869789063261851od_o_o @ ( max_ex7572203241849361634od_o_o @ S ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( max_ex7572203241849361634od_o_o @ R ) ) ) ).
% max_ext_compat
thf(fact_1097_max__ext__compat,axiom,
! [R: set_Pr3205448351804440111at_nat,S: set_Pr3205448351804440111at_nat] :
( ( ord_le559745420748760527at_nat @ ( relcom5008999610213910632at_nat @ R @ S ) @ R )
=> ( ord_le7221101089583142715at_nat @ ( relcom4220097841396419978at_nat @ ( max_ex5394131240172625916at_nat @ R ) @ ( sup_su8834989330170994543at_nat @ ( max_ex5394131240172625916at_nat @ S ) @ ( insert1929353307917009419at_nat @ ( produc4461856759730394867at_nat @ bot_bo1973934853101755969at_nat @ bot_bo1973934853101755969at_nat ) @ bot_bo785540007316762631at_nat ) ) ) @ ( max_ex5394131240172625916at_nat @ R ) ) ) ).
% max_ext_compat
thf(fact_1098_rtrancl__image__advance,axiom,
! [Q2: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat,Q0: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Q2 @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ Q0 ) )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ X4 ) @ R )
=> ( member6956540099943067662at_nat @ X4 @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1099_rtrancl__image__advance,axiom,
! [Q2: product_prod_o_o,R: set_Pr7011910401827753159od_o_o,Q0: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Q2 @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ Q0 ) )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ X4 ) @ R )
=> ( member7466972457876170832od_o_o @ X4 @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1100_rtrancl__image__advance,axiom,
! [Q2: nat,R: set_Pr1261947904930325089at_nat,Q0: set_nat,X4: nat] :
( ( member_nat @ Q2 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ Q0 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ X4 ) @ R )
=> ( member_nat @ X4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1101_rtrancl__image__advance,axiom,
! [Q2: $o,R: set_Product_prod_o_o,Q0: set_o,X4: $o] :
( ( member_o @ Q2 @ ( image_o_o @ ( transitive_rtrancl_o @ R ) @ Q0 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ X4 ) @ R )
=> ( member_o @ X4 @ ( image_o_o @ ( transitive_rtrancl_o @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1102_rtrancl__image__advance__rtrancl,axiom,
! [Q2: produc5825016348098550007at_nat,R: set_Pr3205448351804440111at_nat,Q0: set_Pr2645174627780777389at_nat,X4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Q2 @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ Q0 ) )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ Q2 @ X4 ) @ ( transi7565848034366790580at_nat @ R ) )
=> ( member6956540099943067662at_nat @ X4 @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1103_rtrancl__image__advance__rtrancl,axiom,
! [Q2: product_prod_o_o,R: set_Pr7011910401827753159od_o_o,Q0: set_Product_prod_o_o,X4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Q2 @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ Q0 ) )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Q2 @ X4 ) @ ( transi4448435615503354666od_o_o @ R ) )
=> ( member7466972457876170832od_o_o @ X4 @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1104_rtrancl__image__advance__rtrancl,axiom,
! [Q2: nat,R: set_Pr1261947904930325089at_nat,Q0: set_nat,X4: nat] :
( ( member_nat @ Q2 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ Q0 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q2 @ X4 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member_nat @ X4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1105_rtrancl__image__advance__rtrancl,axiom,
! [Q2: $o,R: set_Product_prod_o_o,Q0: set_o,X4: $o] :
( ( member_o @ Q2 @ ( image_o_o @ ( transitive_rtrancl_o @ R ) @ Q0 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q2 @ X4 ) @ ( transitive_rtrancl_o @ R ) )
=> ( member_o @ X4 @ ( image_o_o @ ( transitive_rtrancl_o @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1106_insert__Image__triv,axiom,
! [X4: relational_fmla_a_b,R3: set_Re381260168593705685la_a_b,Y3: set_Pr1261947904930325089at_nat,Qeq2: set_Pr2645174627780777389at_nat] :
( ~ ( member4680049679412964150la_a_b @ X4 @ R3 )
=> ( ( image_2866438327291862765at_nat @ ( insert3260075854425521959at_nat @ ( produc760948067230524457at_nat @ X4 @ Y3 ) @ Qeq2 ) @ R3 )
= ( image_2866438327291862765at_nat @ Qeq2 @ R3 ) ) ) ).
% insert_Image_triv
thf(fact_1107_insert__Image__triv,axiom,
! [X4: $o,R3: set_o,Y3: $o,Qeq2: set_Product_prod_o_o] :
( ~ ( member_o @ X4 @ R3 )
=> ( ( image_o_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ Qeq2 ) @ R3 )
= ( image_o_o @ Qeq2 @ R3 ) ) ) ).
% insert_Image_triv
thf(fact_1108_union__comp__emptyL,axiom,
! [A4: set_Product_prod_o_o,C2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( relcomp_o_o_o @ A4 @ C2 )
= bot_bo7073875226086086771od_o_o )
=> ( ( ( relcomp_o_o_o @ B5 @ C2 )
= bot_bo7073875226086086771od_o_o )
=> ( ( relcomp_o_o_o @ ( sup_su5769328420594410459od_o_o @ A4 @ B5 ) @ C2 )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% union_comp_emptyL
thf(fact_1109_union__comp__emptyR,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,C2: set_Product_prod_o_o] :
( ( ( relcomp_o_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o )
=> ( ( ( relcomp_o_o_o @ A4 @ C2 )
= bot_bo7073875226086086771od_o_o )
=> ( ( relcomp_o_o_o @ A4 @ ( sup_su5769328420594410459od_o_o @ B5 @ C2 ) )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% union_comp_emptyR
thf(fact_1110_finite__Image,axiom,
! [R: set_Pr2645174627780777389at_nat,A4: set_Re381260168593705685la_a_b] :
( ( finite6790245451575510286at_nat @ R )
=> ( finite9047747110432174090at_nat @ ( image_2866438327291862765at_nat @ R @ A4 ) ) ) ).
% finite_Image
thf(fact_1111_partial__order__on__relation__ofI,axiom,
! [A4: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > produc5825016348098550007at_nat > $o] :
( ! [A2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( P2 @ A2 @ A2 ) )
=> ( ! [A2: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat,C4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ A4 )
=> ( ( member6956540099943067662at_nat @ C4 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ C4 )
=> ( P2 @ A2 @ C4 ) ) ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) )
=> ( order_7623158883673130818at_nat @ A4 @ ( order_7782703812481248809at_nat @ P2 @ A4 ) ) ) ) ) ).
% partial_order_on_relation_ofI
thf(fact_1112_partial__order__on__relation__ofI,axiom,
! [A4: set_Product_prod_o_o,P2: product_prod_o_o > product_prod_o_o > $o] :
( ! [A2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( P2 @ A2 @ A2 ) )
=> ( ! [A2: product_prod_o_o,B2: product_prod_o_o,C4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ C4 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ C4 )
=> ( P2 @ A2 @ C4 ) ) ) ) ) )
=> ( ! [A2: product_prod_o_o,B2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) )
=> ( order_4683604368562820124od_o_o @ A4 @ ( order_4515543550419992437od_o_o @ P2 @ A4 ) ) ) ) ) ).
% partial_order_on_relation_ofI
thf(fact_1113_partial__order__on__relation__ofI,axiom,
! [A4: set_nat,P2: nat > nat > $o] :
( ! [A2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( P2 @ A2 @ A2 ) )
=> ( ! [A2: nat,B2: nat,C4: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ A4 )
=> ( ( member_nat @ C4 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ C4 )
=> ( P2 @ A2 @ C4 ) ) ) ) ) )
=> ( ! [A2: nat,B2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ A4 )
=> ( ( P2 @ A2 @ B2 )
=> ( ( P2 @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) )
=> ( order_5251275573222108571on_nat @ A4 @ ( order_1962693441708344834of_nat @ P2 @ A4 ) ) ) ) ) ).
% partial_order_on_relation_ofI
thf(fact_1114_br__comp__alt_H,axiom,
! [Alpha: relational_fmla_a_b > relational_fmla_a_b,I: relational_fmla_a_b > $o,R: set_Pr2645174627780777389at_nat] :
( ( relcom1330972924984534752at_nat
@ ( collec4721071812894608138la_a_b
@ ( produc4506552522083660972_a_b_o
@ ^ [C3: relational_fmla_a_b,A3: relational_fmla_a_b] :
( ( A3
= ( Alpha @ C3 ) )
& ( I @ C3 ) ) ) )
@ R )
= ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [C3: relational_fmla_a_b,A3: set_Pr1261947904930325089at_nat] :
( ( I @ C3 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ ( Alpha @ C3 ) @ A3 ) @ R ) ) ) ) ) ).
% br_comp_alt'
thf(fact_1115_br__comp__alt_H,axiom,
! [Alpha: relational_fmla_a_b > set_Pr1261947904930325089at_nat,I: relational_fmla_a_b > $o,R: set_Pr4329608150637261639at_nat] :
( ( relcom1463679224352411744at_nat
@ ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [C3: relational_fmla_a_b,A3: set_Pr1261947904930325089at_nat] :
( ( A3
= ( Alpha @ C3 ) )
& ( I @ C3 ) ) ) )
@ R )
= ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [C3: relational_fmla_a_b,A3: set_Pr1261947904930325089at_nat] :
( ( I @ C3 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( Alpha @ C3 ) @ A3 ) @ R ) ) ) ) ) ).
% br_comp_alt'
thf(fact_1116_br__comp__alt_H,axiom,
! [Alpha: $o > $o,I: $o > $o,R: set_Product_prod_o_o] :
( ( relcomp_o_o_o
@ ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [C3: $o,A3: $o] :
( ( A3
= ( Alpha @ C3 ) )
& ( I @ C3 ) ) ) )
@ R )
= ( collec3167064739498627218od_o_o
@ ( produc6197397395684419436_o_o_o
@ ^ [C3: $o,A3: $o] :
( ( I @ C3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( Alpha @ C3 ) @ A3 ) @ R ) ) ) ) ) ).
% br_comp_alt'
thf(fact_1117_Image__def,axiom,
( image_2866438327291862765at_nat
= ( ^ [R2: set_Pr2645174627780777389at_nat,S2: set_Re381260168593705685la_a_b] :
( collec5514110066124741708at_nat
@ ^ [Y4: set_Pr1261947904930325089at_nat] :
? [X: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ S2 )
& ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% Image_def
thf(fact_1118_Image__def,axiom,
( image_o_o
= ( ^ [R2: set_Product_prod_o_o,S2: set_o] :
( collect_o
@ ^ [Y4: $o] :
? [X: $o] :
( ( member_o @ X @ S2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% Image_def
thf(fact_1119_Image__empty__rtrancl__Image__id,axiom,
! [R: set_Pr1261947904930325089at_nat,V: nat] :
( ( ( image_nat_nat @ R @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1120_Image__empty__rtrancl__Image__id,axiom,
! [R: set_Pr7011910401827753159od_o_o,V: product_prod_o_o] :
( ( ( image_539601260427297683od_o_o @ R @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) )
= bot_bo7073875226086086771od_o_o )
=> ( ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) )
= ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1121_Image__empty__rtrancl__Image__id,axiom,
! [R: set_Pr3205448351804440111at_nat,V: produc5825016348098550007at_nat] :
( ( ( image_4491765980459089603at_nat @ R @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) )
= bot_bo1973934853101755969at_nat )
=> ( ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) )
= ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1122_Image__empty__trancl__Image__empty,axiom,
! [R: set_Pr1261947904930325089at_nat,V: nat] :
( ( ( image_nat_nat @ R @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_1123_Image__empty__trancl__Image__empty,axiom,
! [R: set_Pr7011910401827753159od_o_o,V: product_prod_o_o] :
( ( ( image_539601260427297683od_o_o @ R @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) )
= bot_bo7073875226086086771od_o_o )
=> ( ( image_539601260427297683od_o_o @ ( transi1494748803203666376od_o_o @ R ) @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) )
= bot_bo7073875226086086771od_o_o ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_1124_Image__empty__trancl__Image__empty,axiom,
! [R: set_Pr3205448351804440111at_nat,V: produc5825016348098550007at_nat] :
( ( ( image_4491765980459089603at_nat @ R @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) )
= bot_bo1973934853101755969at_nat )
=> ( ( image_4491765980459089603at_nat @ ( transi5629900152392516758at_nat @ R ) @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) )
= bot_bo1973934853101755969at_nat ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_1125_partial__order__on__empty,axiom,
order_8538205470752950541r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% partial_order_on_empty
thf(fact_1126_partial__order__on__empty,axiom,
order_5251275573222108571on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% partial_order_on_empty
thf(fact_1127_partial__order__on__empty,axiom,
order_4683604368562820124od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo2881837355931550515od_o_o ).
% partial_order_on_empty
thf(fact_1128_partial__order__on__empty,axiom,
order_7623158883673130818at_nat @ bot_bo1973934853101755969at_nat @ bot_bo2147731989722242715at_nat ).
% partial_order_on_empty
thf(fact_1129_finite__rtrancl__Image,axiom,
! [R: set_Pr3205448351804440111at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( finite8185985059073859960at_nat @ R )
=> ( ( finite6790245451575510286at_nat @ A4 )
=> ( finite6790245451575510286at_nat @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ A4 ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_1130_Image__singleton,axiom,
! [R3: set_Pr2645174627780777389at_nat,A: relational_fmla_a_b] :
( ( image_2866438327291862765at_nat @ R3 @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
= ( collec5514110066124741708at_nat
@ ^ [B3: set_Pr1261947904930325089at_nat] : ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ A @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1131_Image__singleton,axiom,
! [R3: set_Product_prod_o_o,A: $o] :
( ( image_o_o @ R3 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( collect_o
@ ^ [B3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1132_Image__singleton,axiom,
! [R3: set_Pr2590682000163020040at_nat,A: nat] :
( ( image_6961426772032973084at_nat @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( collec7697611494764367948at_nat
@ ^ [B3: produc5825016348098550007at_nat] : ( member8437817037223515089at_nat @ ( produc3203310505889212640at_nat @ A @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1133_Image__singleton,axiom,
! [R3: set_Pr2403587099507754371at_nat,A: product_prod_o_o] :
( ( image_5920465547893864843at_nat @ R3 @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
= ( collec7697611494764367948at_nat
@ ^ [B3: produc5825016348098550007at_nat] : ( member4819239714442960612at_nat @ ( produc8174103719594181319at_nat @ A @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1134_Image__singleton,axiom,
! [R3: set_Pr3205448351804440111at_nat,A: produc5825016348098550007at_nat] :
( ( image_4491765980459089603at_nat @ R3 @ ( insert3260075854425521959at_nat @ A @ bot_bo1973934853101755969at_nat ) )
= ( collec7697611494764367948at_nat
@ ^ [B3: produc5825016348098550007at_nat] : ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ A @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1135_rtrancl__apply__insert,axiom,
! [R: set_Pr1261947904930325089at_nat,X4: nat,S: set_nat] :
( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( insert_nat @ X4 @ S ) )
= ( insert_nat @ X4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( sup_sup_set_nat @ S @ ( image_nat_nat @ R @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1136_rtrancl__apply__insert,axiom,
! [R: set_Pr7011910401827753159od_o_o,X4: product_prod_o_o,S: set_Product_prod_o_o] :
( ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ ( insert6201435330877294327od_o_o @ X4 @ S ) )
= ( insert6201435330877294327od_o_o @ X4 @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ R ) @ ( sup_su5769328420594410459od_o_o @ S @ ( image_539601260427297683od_o_o @ R @ ( insert6201435330877294327od_o_o @ X4 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1137_rtrancl__apply__insert,axiom,
! [R: set_Pr3205448351804440111at_nat,X4: produc5825016348098550007at_nat,S: set_Pr2645174627780777389at_nat] :
( ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ ( insert3260075854425521959at_nat @ X4 @ S ) )
= ( insert3260075854425521959at_nat @ X4 @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ R ) @ ( sup_su6099978769595272409at_nat @ S @ ( image_4491765980459089603at_nat @ R @ ( insert3260075854425521959at_nat @ X4 @ bot_bo1973934853101755969at_nat ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1138_E__closed__restr__reach__cases,axiom,
! [U: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ ( transi7565848034366790580at_nat @ E ) )
=> ( ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ E @ R ) @ R )
=> ( ~ ( member6956540099943067662at_nat @ V @ R )
=> ~ ( ~ ( member6956540099943067662at_nat @ U @ R )
=> ~ ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ ( transi7565848034366790580at_nat @ ( rel_re178346425959758229at_nat @ E @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_1139_E__closed__restr__reach__cases,axiom,
! [U: product_prod_o_o,V: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ ( transi4448435615503354666od_o_o @ E ) )
=> ( ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ E @ R ) @ R )
=> ( ~ ( member7466972457876170832od_o_o @ V @ R )
=> ~ ( ~ ( member7466972457876170832od_o_o @ U @ R )
=> ~ ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ ( transi4448435615503354666od_o_o @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_1140_E__closed__restr__reach__cases,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R ) @ R )
=> ( ~ ( member_nat @ V @ R )
=> ~ ( ~ ( member_nat @ U @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ ( rel_restrict_nat @ E @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_1141_E__closed__restr__reach__cases,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ E ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R ) @ R )
=> ( ~ ( member_o @ V @ R )
=> ~ ( ~ ( member_o @ U @ R )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ ( rel_restrict_o @ E @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_1142_rel__restrict__tranclI,axiom,
! [X4: produc5825016348098550007at_nat,Y3: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,R: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( transi5629900152392516758at_nat @ E ) )
=> ( ~ ( member6956540099943067662at_nat @ X4 @ R )
=> ( ~ ( member6956540099943067662at_nat @ Y3 @ R )
=> ( ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ E @ R ) @ R )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ X4 @ Y3 ) @ ( transi5629900152392516758at_nat @ ( rel_re178346425959758229at_nat @ E @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1143_rel__restrict__tranclI,axiom,
! [X4: product_prod_o_o,Y3: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,R: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( transi1494748803203666376od_o_o @ E ) )
=> ( ~ ( member7466972457876170832od_o_o @ X4 @ R )
=> ( ~ ( member7466972457876170832od_o_o @ Y3 @ R )
=> ( ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ E @ R ) @ R )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X4 @ Y3 ) @ ( transi1494748803203666376od_o_o @ ( rel_re6977384794829015305od_o_o @ E @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1144_rel__restrict__tranclI,axiom,
! [X4: nat,Y3: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( transi6264000038957366511cl_nat @ E ) )
=> ( ~ ( member_nat @ X4 @ R )
=> ( ~ ( member_nat @ Y3 @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1145_rel__restrict__tranclI,axiom,
! [X4: $o,Y3: $o,E: set_Product_prod_o_o,R: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ E ) )
=> ( ~ ( member_o @ X4 @ R )
=> ( ~ ( member_o @ Y3 @ R )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R ) @ R )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1146_subset__Image__Image__iff,axiom,
! [R3: set_Product_prod_o_o,A4: set_o,B5: set_o] :
( ( order_preorder_on_o @ ( field_o @ R3 ) @ R3 )
=> ( ( ord_less_eq_set_o @ A4 @ ( field_o @ R3 ) )
=> ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R3 ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R3 @ A4 ) @ ( image_o_o @ R3 @ B5 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ? [Y4: $o] :
( ( member_o @ Y4 @ B5 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1147_rtrancl__Image__advance__ss,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
=> ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1148_rtrancl__Image__advance__ss,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1149_rtrancl__Image__advance__ss,axiom,
! [U: product_prod_o_o,V: product_prod_o_o,E: set_Pr7011910401827753159od_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ E )
=> ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) ) @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ U @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1150_rtrancl__Image__advance__ss,axiom,
! [U: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ E )
=> ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ E ) @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) ) @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ E ) @ ( insert3260075854425521959at_nat @ U @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1151_finite__reachable__advance,axiom,
! [E: set_Product_prod_o_o,V0: $o,V: $o] :
( ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V0 @ bot_bot_set_o ) ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V0 @ V ) @ ( transitive_rtrancl_o @ E ) )
=> ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1152_finite__reachable__advance,axiom,
! [E: set_Pr1261947904930325089at_nat,V0: nat,V: nat] :
( ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V0 @ bot_bot_set_nat ) ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V0 @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1153_finite__reachable__advance,axiom,
! [E: set_Pr7011910401827753159od_o_o,V0: product_prod_o_o,V: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ V0 @ bot_bo7073875226086086771od_o_o ) ) )
=> ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ V0 @ V ) @ ( transi4448435615503354666od_o_o @ E ) )
=> ( finite6120865539452801872od_o_o @ ( image_539601260427297683od_o_o @ ( transi4448435615503354666od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1154_finite__reachable__advance,axiom,
! [E: set_Pr3205448351804440111at_nat,V0: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat] :
( ( finite6790245451575510286at_nat @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ E ) @ ( insert3260075854425521959at_nat @ V0 @ bot_bo1973934853101755969at_nat ) ) )
=> ( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ V0 @ V ) @ ( transi7565848034366790580at_nat @ E ) )
=> ( finite6790245451575510286at_nat @ ( image_4491765980459089603at_nat @ ( transi7565848034366790580at_nat @ E ) @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1155_trancl__Image__advance__ss,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
=> ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_1156_trancl__Image__advance__ss,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_1157_trancl__Image__advance__ss,axiom,
! [U: product_prod_o_o,V: product_prod_o_o,E: set_Pr7011910401827753159od_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ E )
=> ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ ( transi1494748803203666376od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ V @ bot_bo7073875226086086771od_o_o ) ) @ ( image_539601260427297683od_o_o @ ( transi1494748803203666376od_o_o @ E ) @ ( insert6201435330877294327od_o_o @ U @ bot_bo7073875226086086771od_o_o ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_1158_trancl__Image__advance__ss,axiom,
! [U: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ E )
=> ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ ( transi5629900152392516758at_nat @ E ) @ ( insert3260075854425521959at_nat @ V @ bot_bo1973934853101755969at_nat ) ) @ ( image_4491765980459089603at_nat @ ( transi5629900152392516758at_nat @ E ) @ ( insert3260075854425521959at_nat @ U @ bot_bo1973934853101755969at_nat ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_1159_min__ext__compat,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R @ S ) @ R )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R ) ) ) ).
% min_ext_compat
thf(fact_1160_min__ext__compat,axiom,
! [R: set_Pr7011910401827753159od_o_o,S: set_Pr7011910401827753159od_o_o] :
( ( ord_le6026832634849756263od_o_o @ ( relcom4901197468039914022od_o_o @ R @ S ) @ R )
=> ( ord_le7586318239878589543od_o_o @ ( relcom8634166700253352710od_o_o @ ( min_ex2812920868796073424od_o_o @ R ) @ ( sup_su4078869789063261851od_o_o @ ( min_ex2812920868796073424od_o_o @ S ) @ ( insert3855699742481506999od_o_o @ ( produc763777882069021527od_o_o @ bot_bo7073875226086086771od_o_o @ bot_bo7073875226086086771od_o_o ) @ bot_bo498831808395348787od_o_o ) ) ) @ ( min_ex2812920868796073424od_o_o @ R ) ) ) ).
% min_ext_compat
thf(fact_1161_min__ext__compat,axiom,
! [R: set_Pr3205448351804440111at_nat,S: set_Pr3205448351804440111at_nat] :
( ( ord_le559745420748760527at_nat @ ( relcom5008999610213910632at_nat @ R @ S ) @ R )
=> ( ord_le7221101089583142715at_nat @ ( relcom4220097841396419978at_nat @ ( min_ex761615834677605006at_nat @ R ) @ ( sup_su8834989330170994543at_nat @ ( min_ex761615834677605006at_nat @ S ) @ ( insert1929353307917009419at_nat @ ( produc4461856759730394867at_nat @ bot_bo1973934853101755969at_nat @ bot_bo1973934853101755969at_nat ) @ bot_bo785540007316762631at_nat ) ) ) @ ( min_ex761615834677605006at_nat @ R ) ) ) ).
% min_ext_compat
thf(fact_1162_trancl__restrict__reachable,axiom,
! [U: produc5825016348098550007at_nat,V: produc5825016348098550007at_nat,E: set_Pr3205448351804440111at_nat,S: set_Pr2645174627780777389at_nat] :
( ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V ) @ ( transi5629900152392516758at_nat @ E ) )
=> ( ( ord_le8520675249591772685at_nat @ ( image_4491765980459089603at_nat @ E @ S ) @ S )
=> ( ( member6956540099943067662at_nat @ U @ S )
=> ( member4689715034089797240at_nat @ ( produc3112276593429206151at_nat @ U @ V )
@ ( transi5629900152392516758at_nat
@ ( inf_in820512908644341533at_nat @ E
@ ( produc984725824077784504at_nat @ S
@ ^ [Uu: produc5825016348098550007at_nat] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_1163_trancl__restrict__reachable,axiom,
! [U: product_prod_o_o,V: product_prod_o_o,E: set_Pr7011910401827753159od_o_o,S: set_Product_prod_o_o] :
( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V ) @ ( transi1494748803203666376od_o_o @ E ) )
=> ( ( ord_le192195778054999463od_o_o @ ( image_539601260427297683od_o_o @ E @ S ) @ S )
=> ( ( member7466972457876170832od_o_o @ U @ S )
=> ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ U @ V )
@ ( transi1494748803203666376od_o_o
@ ( inf_in2119557797576608949od_o_o @ E
@ ( produc6514336722952383752od_o_o @ S
@ ^ [Uu: product_prod_o_o] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_1164_trancl__restrict__reachable,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,S: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi6264000038957366511cl_nat @ E ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ S ) @ S )
=> ( ( member_nat @ U @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V )
@ ( transi6264000038957366511cl_nat
@ ( inf_in2572325071724192079at_nat @ E
@ ( produc457027306803732586at_nat @ S
@ ^ [Uu: nat] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_1165_trancl__restrict__reachable,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o,S: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_trancl_o @ E ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ E @ S ) @ S )
=> ( ( member_o @ U @ S )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V )
@ ( transitive_trancl_o
@ ( inf_in4898592226082374645od_o_o @ E
@ ( product_Sigma_o_o @ S
@ ^ [Uu: $o] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_1166_Id__on__fold,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( finite6790245451575510286at_nat @ A4 )
=> ( ( id_on_89751120774077850at_nat @ A4 )
= ( finite7471822932415046446at_nat
@ ^ [X: produc5825016348098550007at_nat] : ( insert622411394103358111at_nat @ ( produc3112276593429206151at_nat @ X @ X ) )
@ bot_bo2147731989722242715at_nat
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_1167_Id__on__fold,axiom,
! [A4: set_o] :
( ( finite_finite_o @ A4 )
=> ( ( id_on_o @ A4 )
= ( finite7504994868313311671od_o_o
@ ^ [X: $o] : ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) )
@ bot_bo7073875226086086771od_o_o
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_1168_Int__iff,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) )
= ( ( member6956540099943067662at_nat @ C @ A4 )
& ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_1169_Int__iff,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) )
= ( ( member7466972457876170832od_o_o @ C @ A4 )
& ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_1170_Int__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ( member_nat @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_1171_IntI,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ A4 )
=> ( ( member6956540099943067662at_nat @ C @ B5 )
=> ( member6956540099943067662at_nat @ C @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_1172_IntI,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ A4 )
=> ( ( member7466972457876170832od_o_o @ C @ B5 )
=> ( member7466972457876170832od_o_o @ C @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_1173_IntI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% IntI
thf(fact_1174_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1175_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ X4 @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1176_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ X4 @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1177_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1178_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ bot_bo7073875226086086771od_o_o @ X4 )
= bot_bo7073875226086086771od_o_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1179_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ bot_bo1973934853101755969at_nat @ X4 )
= bot_bo1973934853101755969at_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1180_inf__bot__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1181_inf__bot__right,axiom,
! [X4: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ X4 @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% inf_bot_right
thf(fact_1182_inf__bot__right,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ X4 @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% inf_bot_right
thf(fact_1183_inf__bot__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1184_inf__bot__left,axiom,
! [X4: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ bot_bo7073875226086086771od_o_o @ X4 )
= bot_bo7073875226086086771od_o_o ) ).
% inf_bot_left
thf(fact_1185_inf__bot__left,axiom,
! [X4: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ bot_bo1973934853101755969at_nat @ X4 )
= bot_bo1973934853101755969at_nat ) ).
% inf_bot_left
thf(fact_1186_Int__insert__left__if0,axiom,
! [A: produc5825016348098550007at_nat,C2: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ C2 )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B5 ) @ C2 )
= ( inf_in8776938414804536127at_nat @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1187_Int__insert__left__if0,axiom,
! [A: product_prod_o_o,C2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ A @ C2 )
=> ( ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ B5 ) @ C2 )
= ( inf_in4898592226082374645od_o_o @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1188_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1189_Int__insert__left__if1,axiom,
! [A: produc5825016348098550007at_nat,C2: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ C2 )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B5 ) @ C2 )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1190_Int__insert__left__if1,axiom,
! [A: product_prod_o_o,C2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ C2 )
=> ( ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ B5 ) @ C2 )
= ( insert6201435330877294327od_o_o @ A @ ( inf_in4898592226082374645od_o_o @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1191_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1192_Int__insert__right__if0,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ~ ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( inf_in8776938414804536127at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1193_Int__insert__right__if0,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ~ ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( inf_in4898592226082374645od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1194_Int__insert__right__if0,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1195_Int__insert__right__if1,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( inf_in8776938414804536127at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1196_Int__insert__right__if1,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( inf_in4898592226082374645od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( insert6201435330877294327od_o_o @ A @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1197_Int__insert__right__if1,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1198_finite__Int,axiom,
! [F4: set_Pr2645174627780777389at_nat,G2: set_Pr2645174627780777389at_nat] :
( ( ( finite6790245451575510286at_nat @ F4 )
| ( finite6790245451575510286at_nat @ G2 ) )
=> ( finite6790245451575510286at_nat @ ( inf_in8776938414804536127at_nat @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_1199_insert__disjoint_I1_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1200_insert__disjoint_I1_J,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ A4 ) @ B5 )
= bot_bo7073875226086086771od_o_o )
= ( ~ ( member7466972457876170832od_o_o @ A @ B5 )
& ( ( inf_in4898592226082374645od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% insert_disjoint(1)
thf(fact_1201_insert__disjoint_I1_J,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ A4 ) @ B5 )
= bot_bo1973934853101755969at_nat )
= ( ~ ( member6956540099943067662at_nat @ A @ B5 )
& ( ( inf_in8776938414804536127at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1202_insert__disjoint_I2_J,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member_nat @ A @ B5 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1203_insert__disjoint_I2_J,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ A4 ) @ B5 ) )
= ( ~ ( member7466972457876170832od_o_o @ A @ B5 )
& ( bot_bo7073875226086086771od_o_o
= ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1204_insert__disjoint_I2_J,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ A4 ) @ B5 ) )
= ( ~ ( member6956540099943067662at_nat @ A @ B5 )
& ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1205_disjoint__insert_I1_J,axiom,
! [B5: set_nat,A: nat,A4: set_nat] :
( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A4 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B5 )
& ( ( inf_inf_set_nat @ B5 @ A4 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1206_disjoint__insert_I1_J,axiom,
! [B5: set_Product_prod_o_o,A: product_prod_o_o,A4: set_Product_prod_o_o] :
( ( ( inf_in4898592226082374645od_o_o @ B5 @ ( insert6201435330877294327od_o_o @ A @ A4 ) )
= bot_bo7073875226086086771od_o_o )
= ( ~ ( member7466972457876170832od_o_o @ A @ B5 )
& ( ( inf_in4898592226082374645od_o_o @ B5 @ A4 )
= bot_bo7073875226086086771od_o_o ) ) ) ).
% disjoint_insert(1)
thf(fact_1207_disjoint__insert_I1_J,axiom,
! [B5: set_Pr2645174627780777389at_nat,A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ B5 @ ( insert3260075854425521959at_nat @ A @ A4 ) )
= bot_bo1973934853101755969at_nat )
= ( ~ ( member6956540099943067662at_nat @ A @ B5 )
& ( ( inf_in8776938414804536127at_nat @ B5 @ A4 )
= bot_bo1973934853101755969at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1208_disjoint__insert_I2_J,axiom,
! [A4: set_nat,B: nat,B5: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) )
= ( ~ ( member_nat @ B @ A4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1209_disjoint__insert_I2_J,axiom,
! [A4: set_Product_prod_o_o,B: product_prod_o_o,B5: set_Product_prod_o_o] :
( ( bot_bo7073875226086086771od_o_o
= ( inf_in4898592226082374645od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ B @ B5 ) ) )
= ( ~ ( member7466972457876170832od_o_o @ B @ A4 )
& ( bot_bo7073875226086086771od_o_o
= ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1210_disjoint__insert_I2_J,axiom,
! [A4: set_Pr2645174627780777389at_nat,B: produc5825016348098550007at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A4 @ ( insert3260075854425521959at_nat @ B @ B5 ) ) )
= ( ~ ( member6956540099943067662at_nat @ B @ A4 )
& ( bot_bo1973934853101755969at_nat
= ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1211_Diff__disjoint,axiom,
! [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1212_Diff__disjoint,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ A4 @ ( minus_7052513330764642158od_o_o @ B5 @ A4 ) )
= bot_bo7073875226086086771od_o_o ) ).
% Diff_disjoint
thf(fact_1213_Diff__disjoint,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A4 @ ( minus_6698950876951835142at_nat @ B5 @ A4 ) )
= bot_bo1973934853101755969at_nat ) ).
% Diff_disjoint
thf(fact_1214_inf__Sup__absorb,axiom,
! [A4: set_Product_prod_o_o,A: product_prod_o_o] :
( ( finite6120865539452801872od_o_o @ A4 )
=> ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( inf_in4843801116672622357od_o_o @ A @ ( lattic8473092939728248030od_o_o @ A4 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1215_inf__Sup__absorb,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1216_disjointI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% disjointI
thf(fact_1217_disjointI,axiom,
! [A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A )
=> ~ ( member7466972457876170832od_o_o @ X3 @ B ) )
=> ( ( inf_in4898592226082374645od_o_o @ A @ B )
= bot_bo7073875226086086771od_o_o ) ) ).
% disjointI
thf(fact_1218_disjointI,axiom,
! [A: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A )
=> ~ ( member6956540099943067662at_nat @ X3 @ B ) )
=> ( ( inf_in8776938414804536127at_nat @ A @ B )
= bot_bo1973934853101755969at_nat ) ) ).
% disjointI
thf(fact_1219_Int__emptyI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member_nat @ X3 @ B5 ) )
=> ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1220_Int__emptyI,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ~ ( member7466972457876170832od_o_o @ X3 @ B5 ) )
=> ( ( inf_in4898592226082374645od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o ) ) ).
% Int_emptyI
thf(fact_1221_Int__emptyI,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ~ ( member6956540099943067662at_nat @ X3 @ B5 ) )
=> ( ( inf_in8776938414804536127at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat ) ) ).
% Int_emptyI
thf(fact_1222_disjoint__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ~ ( member_nat @ X @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1223_disjoint__iff,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( inf_in4898592226082374645od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
=> ~ ( member7466972457876170832od_o_o @ X @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1224_disjoint__iff,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
=> ~ ( member6956540099943067662at_nat @ X @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_1225_Int__empty__left,axiom,
! [B5: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B5 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1226_Int__empty__left,axiom,
! [B5: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ bot_bo7073875226086086771od_o_o @ B5 )
= bot_bo7073875226086086771od_o_o ) ).
% Int_empty_left
thf(fact_1227_Int__empty__left,axiom,
! [B5: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ bot_bo1973934853101755969at_nat @ B5 )
= bot_bo1973934853101755969at_nat ) ).
% Int_empty_left
thf(fact_1228_Int__empty__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1229_Int__empty__right,axiom,
! [A4: set_Product_prod_o_o] :
( ( inf_in4898592226082374645od_o_o @ A4 @ bot_bo7073875226086086771od_o_o )
= bot_bo7073875226086086771od_o_o ) ).
% Int_empty_right
thf(fact_1230_Int__empty__right,axiom,
! [A4: set_Pr2645174627780777389at_nat] :
( ( inf_in8776938414804536127at_nat @ A4 @ bot_bo1973934853101755969at_nat )
= bot_bo1973934853101755969at_nat ) ).
% Int_empty_right
thf(fact_1231_disjoint__iff__not__equal,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B5 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1232_disjoint__iff__not__equal,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( inf_in4898592226082374645od_o_o @ A4 @ B5 )
= bot_bo7073875226086086771od_o_o )
= ( ! [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A4 )
=> ! [Y4: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ Y4 @ B5 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1233_disjoint__iff__not__equal,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( inf_in8776938414804536127at_nat @ A4 @ B5 )
= bot_bo1973934853101755969at_nat )
= ( ! [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A4 )
=> ! [Y4: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ Y4 @ B5 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1234_Int__insert__left,axiom,
! [A: produc5825016348098550007at_nat,C2: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ A @ C2 )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B5 ) @ C2 )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ B5 @ C2 ) ) ) )
& ( ~ ( member6956540099943067662at_nat @ A @ C2 )
=> ( ( inf_in8776938414804536127at_nat @ ( insert3260075854425521959at_nat @ A @ B5 ) @ C2 )
= ( inf_in8776938414804536127at_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1235_Int__insert__left,axiom,
! [A: product_prod_o_o,C2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ A @ C2 )
=> ( ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ B5 ) @ C2 )
= ( insert6201435330877294327od_o_o @ A @ ( inf_in4898592226082374645od_o_o @ B5 @ C2 ) ) ) )
& ( ~ ( member7466972457876170832od_o_o @ A @ C2 )
=> ( ( inf_in4898592226082374645od_o_o @ ( insert6201435330877294327od_o_o @ A @ B5 ) @ C2 )
= ( inf_in4898592226082374645od_o_o @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1236_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C2 )
= ( inf_inf_set_nat @ B5 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1237_Int__insert__right,axiom,
! [A: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( inf_in8776938414804536127at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( insert3260075854425521959at_nat @ A @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member6956540099943067662at_nat @ A @ A4 )
=> ( ( inf_in8776938414804536127at_nat @ A4 @ ( insert3260075854425521959at_nat @ A @ B5 ) )
= ( inf_in8776938414804536127at_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1238_Int__insert__right,axiom,
! [A: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( inf_in4898592226082374645od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( insert6201435330877294327od_o_o @ A @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) )
& ( ~ ( member7466972457876170832od_o_o @ A @ A4 )
=> ( ( inf_in4898592226082374645od_o_o @ A4 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
= ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1239_Int__insert__right,axiom,
! [A: nat,A4: set_nat,B5: set_nat] :
( ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1240_fold__closed__eq,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,F: produc5825016348098550007at_nat > produc5825016348098550007at_nat > produc5825016348098550007at_nat,G: produc5825016348098550007at_nat > produc5825016348098550007at_nat > produc5825016348098550007at_nat,Z: produc5825016348098550007at_nat] :
( ! [A2: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat,B2: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( member6956540099943067662at_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member6956540099943067662at_nat @ Z @ B5 )
=> ( ( finite1371395723793962296at_nat @ F @ Z @ A4 )
= ( finite1371395723793962296at_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1241_fold__closed__eq,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Product_prod_o_o,F: produc5825016348098550007at_nat > product_prod_o_o > product_prod_o_o,G: produc5825016348098550007at_nat > product_prod_o_o > product_prod_o_o,Z: product_prod_o_o] :
( ! [A2: produc5825016348098550007at_nat,B2: product_prod_o_o] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat,B2: product_prod_o_o] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( member7466972457876170832od_o_o @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member7466972457876170832od_o_o @ Z @ B5 )
=> ( ( finite4263794960621469862od_o_o @ F @ Z @ A4 )
= ( finite4263794960621469862od_o_o @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1242_fold__closed__eq,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_nat,F: produc5825016348098550007at_nat > nat > nat,G: produc5825016348098550007at_nat > nat > nat,Z: nat] :
( ! [A2: produc5825016348098550007at_nat,B2: nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: produc5825016348098550007at_nat,B2: nat] :
( ( member6956540099943067662at_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( member_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member_nat @ Z @ B5 )
=> ( ( finite5936458759689831057at_nat @ F @ Z @ A4 )
= ( finite5936458759689831057at_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1243_fold__closed__eq,axiom,
! [A4: set_Product_prod_o_o,B5: set_Pr2645174627780777389at_nat,F: product_prod_o_o > produc5825016348098550007at_nat > produc5825016348098550007at_nat,G: product_prod_o_o > produc5825016348098550007at_nat > produc5825016348098550007at_nat,Z: produc5825016348098550007at_nat] :
( ! [A2: product_prod_o_o,B2: produc5825016348098550007at_nat] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: product_prod_o_o,B2: produc5825016348098550007at_nat] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( member6956540099943067662at_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member6956540099943067662at_nat @ Z @ B5 )
=> ( ( finite5391307545077869014at_nat @ F @ Z @ A4 )
= ( finite5391307545077869014at_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1244_fold__closed__eq,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,F: product_prod_o_o > product_prod_o_o > product_prod_o_o,G: product_prod_o_o > product_prod_o_o > product_prod_o_o,Z: product_prod_o_o] :
( ! [A2: product_prod_o_o,B2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: product_prod_o_o,B2: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( member7466972457876170832od_o_o @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member7466972457876170832od_o_o @ Z @ B5 )
=> ( ( finite9200831542336347272od_o_o @ F @ Z @ A4 )
= ( finite9200831542336347272od_o_o @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1245_fold__closed__eq,axiom,
! [A4: set_Product_prod_o_o,B5: set_nat,F: product_prod_o_o > nat > nat,G: product_prod_o_o > nat > nat,Z: nat] :
( ! [A2: product_prod_o_o,B2: nat] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: product_prod_o_o,B2: nat] :
( ( member7466972457876170832od_o_o @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( member_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member_nat @ Z @ B5 )
=> ( ( finite7505512383721325103_o_nat @ F @ Z @ A4 )
= ( finite7505512383721325103_o_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1246_fold__closed__eq,axiom,
! [A4: set_nat,B5: set_Pr2645174627780777389at_nat,F: nat > produc5825016348098550007at_nat > produc5825016348098550007at_nat,G: nat > produc5825016348098550007at_nat > produc5825016348098550007at_nat,Z: produc5825016348098550007at_nat] :
( ! [A2: nat,B2: produc5825016348098550007at_nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: nat,B2: produc5825016348098550007at_nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member6956540099943067662at_nat @ B2 @ B5 )
=> ( member6956540099943067662at_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member6956540099943067662at_nat @ Z @ B5 )
=> ( ( finite8214511846341882897at_nat @ F @ Z @ A4 )
= ( finite8214511846341882897at_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1247_fold__closed__eq,axiom,
! [A4: set_nat,B5: set_Product_prod_o_o,F: nat > product_prod_o_o > product_prod_o_o,G: nat > product_prod_o_o > product_prod_o_o,Z: product_prod_o_o] :
( ! [A2: nat,B2: product_prod_o_o] :
( ( member_nat @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: nat,B2: product_prod_o_o] :
( ( member_nat @ A2 @ A4 )
=> ( ( member7466972457876170832od_o_o @ B2 @ B5 )
=> ( member7466972457876170832od_o_o @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member7466972457876170832od_o_o @ Z @ B5 )
=> ( ( finite3789750807609040781od_o_o @ F @ Z @ A4 )
= ( finite3789750807609040781od_o_o @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1248_fold__closed__eq,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z: nat] :
( ! [A2: nat,B2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( ( F @ A2 @ B2 )
= ( G @ A2 @ B2 ) ) ) )
=> ( ! [A2: nat,B2: nat] :
( ( member_nat @ A2 @ A4 )
=> ( ( member_nat @ B2 @ B5 )
=> ( member_nat @ ( G @ A2 @ B2 ) @ B5 ) ) )
=> ( ( member_nat @ Z @ B5 )
=> ( ( finite_fold_nat_nat @ F @ Z @ A4 )
= ( finite_fold_nat_nat @ G @ Z @ A4 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1249_IntD2,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) )
=> ( member6956540099943067662at_nat @ C @ B5 ) ) ).
% IntD2
thf(fact_1250_IntD2,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) )
=> ( member7466972457876170832od_o_o @ C @ B5 ) ) ).
% IntD2
thf(fact_1251_IntD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ B5 ) ) ).
% IntD2
thf(fact_1252_IntD1,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) )
=> ( member6956540099943067662at_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_1253_IntD1,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) )
=> ( member7466972457876170832od_o_o @ C @ A4 ) ) ).
% IntD1
thf(fact_1254_IntD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_1255_IntE,axiom,
! [C: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat] :
( ( member6956540099943067662at_nat @ C @ ( inf_in8776938414804536127at_nat @ A4 @ B5 ) )
=> ~ ( ( member6956540099943067662at_nat @ C @ A4 )
=> ~ ( member6956540099943067662at_nat @ C @ B5 ) ) ) ).
% IntE
thf(fact_1256_IntE,axiom,
! [C: product_prod_o_o,A4: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ C @ ( inf_in4898592226082374645od_o_o @ A4 @ B5 ) )
=> ~ ( ( member7466972457876170832od_o_o @ C @ A4 )
=> ~ ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).
% IntE
thf(fact_1257_IntE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ~ ( member_nat @ C @ B5 ) ) ) ).
% IntE
thf(fact_1258_Int__def,axiom,
( inf_in4898592226082374645od_o_o
= ( ^ [A7: set_Product_prod_o_o,B7: set_Product_prod_o_o] :
( collec3167064739498627218od_o_o
@ ^ [X: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X @ A7 )
& ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_1259_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A7: set_nat,B7: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A7 )
& ( member_nat @ X @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_1260_Int__def,axiom,
( inf_in8776938414804536127at_nat
= ( ^ [A7: set_Pr2645174627780777389at_nat,B7: set_Pr2645174627780777389at_nat] :
( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X @ A7 )
& ( member6956540099943067662at_nat @ X @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_1261_Int__Collect,axiom,
! [X4: product_prod_o_o,A4: set_Product_prod_o_o,P2: product_prod_o_o > $o] :
( ( member7466972457876170832od_o_o @ X4 @ ( inf_in4898592226082374645od_o_o @ A4 @ ( collec3167064739498627218od_o_o @ P2 ) ) )
= ( ( member7466972457876170832od_o_o @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_1262_Int__Collect,axiom,
! [X4: nat,A4: set_nat,P2: nat > $o] :
( ( member_nat @ X4 @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_1263_Int__Collect,axiom,
! [X4: produc5825016348098550007at_nat,A4: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o] :
( ( member6956540099943067662at_nat @ X4 @ ( inf_in8776938414804536127at_nat @ A4 @ ( collec7697611494764367948at_nat @ P2 ) ) )
= ( ( member6956540099943067662at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_1264_Collect__conj__eq,axiom,
! [P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( collec7697611494764367948at_nat
@ ^ [X: produc5825016348098550007at_nat] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_in8776938414804536127at_nat @ ( collec7697611494764367948at_nat @ P2 ) @ ( collec7697611494764367948at_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1265_Int__Collect__mono,axiom,
! [A4: set_Product_prod_o_o,B5: set_Product_prod_o_o,P2: product_prod_o_o > $o,Q: product_prod_o_o > $o] :
( ( ord_le192195778054999463od_o_o @ A4 @ B5 )
=> ( ! [X3: product_prod_o_o] :
( ( member7466972457876170832od_o_o @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le192195778054999463od_o_o @ ( inf_in4898592226082374645od_o_o @ A4 @ ( collec3167064739498627218od_o_o @ P2 ) ) @ ( inf_in4898592226082374645od_o_o @ B5 @ ( collec3167064739498627218od_o_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1266_Int__Collect__mono,axiom,
! [A4: set_nat,B5: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B5 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1267_Int__Collect__mono,axiom,
! [A4: set_Pr2645174627780777389at_nat,B5: set_Pr2645174627780777389at_nat,P2: produc5825016348098550007at_nat > $o,Q: produc5825016348098550007at_nat > $o] :
( ( ord_le8520675249591772685at_nat @ A4 @ B5 )
=> ( ! [X3: produc5825016348098550007at_nat] :
( ( member6956540099943067662at_nat @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8520675249591772685at_nat @ ( inf_in8776938414804536127at_nat @ A4 @ ( collec7697611494764367948at_nat @ P2 ) ) @ ( inf_in8776938414804536127at_nat @ B5 @ ( collec7697611494764367948at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1268_disjoint__mono,axiom,
! [A: set_Pr2645174627780777389at_nat,A5: set_Pr2645174627780777389at_nat,B: set_Pr2645174627780777389at_nat,B4: set_Pr2645174627780777389at_nat] :
( ( ord_le8520675249591772685at_nat @ A @ A5 )
=> ( ( ord_le8520675249591772685at_nat @ B @ B4 )
=> ( ( ( inf_in8776938414804536127at_nat @ A5 @ B4 )
= bot_bo1973934853101755969at_nat )
=> ( ( inf_in8776938414804536127at_nat @ A @ B )
= bot_bo1973934853101755969at_nat ) ) ) ) ).
% disjoint_mono
thf(fact_1269_param__and__cong2,axiom,
! [A: $o,A5: $o,B: $o,B4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A5 ) @ id_o )
=> ( ( A
=> ( A5
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ B4 ) @ id_o ) ) )
=> ( member7466972457876170832od_o_o
@ ( product_Pair_o_o
@ ( B
& A )
@ ( B4
& A5 ) )
@ id_o ) ) ) ).
% param_and_cong2
thf(fact_1270_param__and__cong1,axiom,
! [A: $o,A5: $o,B: $o,B4: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A5 ) @ id_o )
=> ( ( A
=> ( A5
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ B4 ) @ id_o ) ) )
=> ( member7466972457876170832od_o_o
@ ( product_Pair_o_o
@ ( A
& B )
@ ( A5
& B4 ) )
@ id_o ) ) ) ).
% param_and_cong1
thf(fact_1271_param__bool_I1_J,axiom,
member7466972457876170832od_o_o @ ( product_Pair_o_o @ $true @ $true ) @ id_o ).
% param_bool(1)
thf(fact_1272_param__bool_I2_J,axiom,
member7466972457876170832od_o_o @ ( product_Pair_o_o @ $false @ $false ) @ id_o ).
% param_bool(2)
thf(fact_1273_less__than__bool__iff,axiom,
! [X4: $o,Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ refine8250873594941227145n_bool )
= ( ~ X4
& Y3 ) ) ).
% less_than_bool_iff
thf(fact_1274_autoref__bool_I1_J,axiom,
! [X4: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ X4 ) @ id_o ) ).
% autoref_bool(1)
% Conjectures (2)
thf(conj_0,hypothesis,
finite6790245451575510286at_nat @ q ).
thf(conj_1,conjecture,
( finite6790245451575510286at_nat
@ ( collec7697611494764367948at_nat
@ ( produc5861013030283631554_nat_o
@ ^ [Qfix: relational_fmla_a_b,Qeq: set_Pr1261947904930325089at_nat] :
( ( member6956540099943067662at_nat @ ( produc760948067230524457at_nat @ Qfix @ Qeq ) @ q )
& ( ( relati62690040636126068ns_a_b @ Qfix )
!= bot_bot_set_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------