TPTP Problem File: SLH0554^1.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 : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_01178_040867__5897240_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1566 ( 406 unt; 287 typ; 0 def)
% Number of atoms : 4467 (1053 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13207 ( 410 ~; 25 |; 471 &;10085 @)
% ( 0 <=>;2216 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 1342 (1342 >; 0 *; 0 +; 0 <<)
% Number of symbols : 261 ( 258 usr; 24 con; 0-4 aty)
% Number of variables : 4357 ( 433 ^;3669 !; 255 ?;4357 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:23.843
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
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% Relevant facts (1275)
thf(fact_0__092_060open_062_092_060And_062n_O_An_A_092_060in_062_A_092_060S_062_Ae_An_092_060close_062,axiom,
! [N: v] : ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ e @ N ) ) ).
% \<open>\<And>n. n \<in> \<S> e n\<close>
thf(fact_1_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_2_Se_H,axiom,
! [X: v] :
( ( ( member_v @ X @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X )
= cc ) )
& ( ~ ( member_v @ X @ cc )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ e @ X ) ) ) ) ).
% Se'
thf(fact_3_Sup__subset__mono,axiom,
! [A: set_set_set_v,B: set_set_set_v] :
( ( ord_le8117609702905084755_set_v @ A @ B )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_4_Sup__subset__mono,axiom,
! [A: set_se2157405561750842759od_v_v,B: set_se2157405561750842759od_v_v] :
( ( ord_le4688432828652561703od_v_v @ A @ B )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ ( comple6800410395631251988od_v_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_5_Sup__subset__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_6_Sup__subset__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_7_Sup__subset__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_8_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_9_UN__ball__bex__simps_I3_J,axiom,
! [A: set_se8455005133513928103od_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
& ? [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_10_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_o,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ? [Y: $o] :
( ( member_o @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_11_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ? [Y: v] :
( ( member_v @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_12_UN__ball__bex__simps_I1_J,axiom,
! [A: set_se8455005133513928103od_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_13_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ! [Y: $o] :
( ( member_o @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_14_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ! [Y: v] :
( ( member_v @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_15_UnionI,axiom,
! [X3: set_set_v,C: set_set_set_v,A: set_v] :
( ( member_set_set_v @ X3 @ C )
=> ( ( member_set_v @ A @ X3 )
=> ( member_set_v @ A @ ( comple5450237519782573632_set_v @ C ) ) ) ) ).
% UnionI
thf(fact_16_UnionI,axiom,
! [X3: set_se8455005133513928103od_v_v,C: set_se2157405561750842759od_v_v,A: set_Product_prod_v_v] :
( ( member5511408251247217616od_v_v @ X3 @ C )
=> ( ( member8406446414694345712od_v_v @ A @ X3 )
=> ( member8406446414694345712od_v_v @ A @ ( comple6800410395631251988od_v_v @ C ) ) ) ) ).
% UnionI
thf(fact_17_UnionI,axiom,
! [X3: set_set_o,C: set_set_set_o,A: set_o] :
( ( member_set_set_o @ X3 @ C )
=> ( ( member_set_o @ A @ X3 )
=> ( member_set_o @ A @ ( comple4436988014476444997_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_18_UnionI,axiom,
! [X3: set_o,C: set_set_o,A: $o] :
( ( member_set_o @ X3 @ C )
=> ( ( member_o @ A @ X3 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_19_UnionI,axiom,
! [X3: set_Product_prod_v_v,C: set_se8455005133513928103od_v_v,A: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C )
=> ( ( member7453568604450474000od_v_v @ A @ X3 )
=> ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) ) ) ) ).
% UnionI
thf(fact_20_UnionI,axiom,
! [X3: set_v,C: set_set_v,A: v] :
( ( member_set_v @ X3 @ C )
=> ( ( member_v @ A @ X3 )
=> ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) ) ) ) ).
% UnionI
thf(fact_21_Union__iff,axiom,
! [A: set_v,C: set_set_set_v] :
( ( member_set_v @ A @ ( comple5450237519782573632_set_v @ C ) )
= ( ? [X2: set_set_v] :
( ( member_set_set_v @ X2 @ C )
& ( member_set_v @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_22_Union__iff,axiom,
! [A: set_Product_prod_v_v,C: set_se2157405561750842759od_v_v] :
( ( member8406446414694345712od_v_v @ A @ ( comple6800410395631251988od_v_v @ C ) )
= ( ? [X2: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X2 @ C )
& ( member8406446414694345712od_v_v @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_23_Union__iff,axiom,
! [A: set_o,C: set_set_set_o] :
( ( member_set_o @ A @ ( comple4436988014476444997_set_o @ C ) )
= ( ? [X2: set_set_o] :
( ( member_set_set_o @ X2 @ C )
& ( member_set_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_24_Union__iff,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C )
& ( member_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_25_Union__iff,axiom,
! [A: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) )
= ( ? [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C )
& ( member7453568604450474000od_v_v @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_26_Union__iff,axiom,
! [A: v,C: set_set_v] :
( ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ C )
& ( member_v @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_27_subsetI,axiom,
! [A: set_set_o,B: set_set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( member_set_o @ X4 @ B ) )
=> ( ord_le4374716579403074808_set_o @ A @ B ) ) ).
% subsetI
thf(fact_28_subsetI,axiom,
! [A: set_v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( member_v @ X4 @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% subsetI
thf(fact_29_subsetI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A )
=> ( member7453568604450474000od_v_v @ X4 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% subsetI
thf(fact_30_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ X4 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_31_subsetI,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( member_set_v @ X4 @ B ) )
=> ( ord_le5216385588623774835_set_v @ A @ B ) ) ).
% subsetI
thf(fact_32_subsetI,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ( member8406446414694345712od_v_v @ X4 @ B ) )
=> ( ord_le4714265922333009223od_v_v @ A @ B ) ) ).
% subsetI
thf(fact_33_subset__antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_34_subset__antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_35_subset__antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_36_subset__antisym,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_37_subset__antisym,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( ord_le4714265922333009223od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_38_order__refl,axiom,
! [X5: set_v] : ( ord_less_eq_set_v @ X5 @ X5 ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X5: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X5 @ X5 ) ).
% order_refl
thf(fact_40_order__refl,axiom,
! [X5: set_o] : ( ord_less_eq_set_o @ X5 @ X5 ) ).
% order_refl
thf(fact_41_order__refl,axiom,
! [X5: $o] : ( ord_less_eq_o @ X5 @ X5 ) ).
% order_refl
thf(fact_42_order__refl,axiom,
! [X5: set_set_v] : ( ord_le5216385588623774835_set_v @ X5 @ X5 ) ).
% order_refl
thf(fact_43_order__refl,axiom,
! [X5: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ X5 @ X5 ) ).
% order_refl
thf(fact_44_dual__order_Orefl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_45_dual__order_Orefl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_46_dual__order_Orefl,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_47_dual__order_Orefl,axiom,
! [A2: $o] : ( ord_less_eq_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_48_dual__order_Orefl,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_49_dual__order_Orefl,axiom,
! [A2: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_50_Union__mono,axiom,
! [A: set_set_set_v,B: set_set_set_v] :
( ( ord_le8117609702905084755_set_v @ A @ B )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Union_mono
thf(fact_51_Union__mono,axiom,
! [A: set_se2157405561750842759od_v_v,B: set_se2157405561750842759od_v_v] :
( ( ord_le4688432828652561703od_v_v @ A @ B )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ ( comple6800410395631251988od_v_v @ B ) ) ) ).
% Union_mono
thf(fact_52_Union__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_mono
thf(fact_53_Union__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_mono
thf(fact_54_Union__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Union_mono
thf(fact_55_less__eq__set__def,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A3: set_set_o,B2: set_set_o] :
( ord_less_eq_set_o_o
@ ^ [X2: set_o] : ( member_set_o @ X2 @ A3 )
@ ^ [X2: set_o] : ( member_set_o @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_56_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
( ord_less_eq_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A3 )
@ ^ [X2: v] : ( member_v @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_57_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_58_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 )
@ ^ [X2: $o] : ( member_o @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_59_less__eq__set__def,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( ord_less_eq_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A3 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_60_less__eq__set__def,axiom,
( ord_le4714265922333009223od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ord_le2647354080893881942_v_v_o
@ ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ A3 )
@ ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_61_order__antisym__conv,axiom,
! [Y2: set_v,X5: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X5 )
=> ( ( ord_less_eq_set_v @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_62_order__antisym__conv,axiom,
! [Y2: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X5 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_63_order__antisym__conv,axiom,
! [Y2: set_o,X5: set_o] :
( ( ord_less_eq_set_o @ Y2 @ X5 )
=> ( ( ord_less_eq_set_o @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_64_order__antisym__conv,axiom,
! [Y2: $o,X5: $o] :
( ( ord_less_eq_o @ Y2 @ X5 )
=> ( ( ord_less_eq_o @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_65_order__antisym__conv,axiom,
! [Y2: set_set_v,X5: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y2 @ X5 )
=> ( ( ord_le5216385588623774835_set_v @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_66_order__antisym__conv,axiom,
! [Y2: set_se8455005133513928103od_v_v,X5: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ Y2 @ X5 )
=> ( ( ord_le4714265922333009223od_v_v @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_67_linorder__le__cases,axiom,
! [X5: $o,Y2: $o] :
( ~ ( ord_less_eq_o @ X5 @ Y2 )
=> ( ord_less_eq_o @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_68_ord__le__eq__subst,axiom,
! [A2: $o,B3: $o,F: $o > $o,C2: $o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A2: set_v,B3: set_v,F: set_v > $o,C2: $o] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A2: set_o,B3: set_o,F: set_o > $o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A2: $o,B3: $o,F: $o > set_v,C2: set_v] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A2: $o,B3: $o,F: $o > set_o,C2: set_o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A2: set_v,B3: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A2: set_v,B3: set_v,F: set_v > set_o,C2: set_o] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A2: set_o,B3: set_o,F: set_o > set_v,C2: set_v] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
! [A2: set_o,B3: set_o,F: set_o > set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
! [A2: $o,B3: $o,F: $o > set_set_v,C2: set_set_v] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__eq__le__subst,axiom,
! [A2: $o,F: $o > $o,B3: $o,C2: $o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A2: $o,F: set_v > $o,B3: set_v,C2: set_v] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A2: $o,F: set_o > $o,B3: set_o,C2: set_o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A2: set_v,F: $o > set_v,B3: $o,C2: $o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A2: set_o,F: $o > set_o,B3: $o,C2: $o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A2: set_v,F: set_v > set_v,B3: set_v,C2: set_v] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A2: set_o,F: set_v > set_o,B3: set_v,C2: set_v] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A2: set_v,F: set_o > set_v,B3: set_o,C2: set_o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A2: set_o,F: set_o > set_o,B3: set_o,C2: set_o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A2: set_set_v,F: $o > set_set_v,B3: $o,C2: $o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_linorder__linear,axiom,
! [X5: $o,Y2: $o] :
( ( ord_less_eq_o @ X5 @ Y2 )
| ( ord_less_eq_o @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_89_order__eq__refl,axiom,
! [X5: set_v,Y2: set_v] :
( ( X5 = Y2 )
=> ( ord_less_eq_set_v @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_90_order__eq__refl,axiom,
! [X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( X5 = Y2 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_91_order__eq__refl,axiom,
! [X5: set_o,Y2: set_o] :
( ( X5 = Y2 )
=> ( ord_less_eq_set_o @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_92_order__eq__refl,axiom,
! [X5: $o,Y2: $o] :
( ( X5 = Y2 )
=> ( ord_less_eq_o @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_93_order__eq__refl,axiom,
! [X5: set_set_v,Y2: set_set_v] :
( ( X5 = Y2 )
=> ( ord_le5216385588623774835_set_v @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_94_order__eq__refl,axiom,
! [X5: set_se8455005133513928103od_v_v,Y2: set_se8455005133513928103od_v_v] :
( ( X5 = Y2 )
=> ( ord_le4714265922333009223od_v_v @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_95_order__subst2,axiom,
! [A2: $o,B3: $o,F: $o > $o,C2: $o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_less_eq_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A2: set_v,B3: set_v,F: set_v > $o,C2: $o] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ord_less_eq_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A2: set_o,B3: set_o,F: set_o > $o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A2: $o,B3: $o,F: $o > set_v,C2: set_v] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A2: $o,B3: $o,F: $o > set_o,C2: set_o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A2: set_v,B3: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A2: set_v,B3: set_v,F: set_v > set_o,C2: set_o] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A2: set_o,B3: set_o,F: set_o > set_v,C2: set_v] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A2: set_o,B3: set_o,F: set_o > set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ ( F @ B3 ) @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A2: $o,B3: $o,F: $o > set_set_v,C2: set_set_v] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_le5216385588623774835_set_v @ ( F @ B3 ) @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst1,axiom,
! [A2: $o,F: $o > $o,B3: $o,C2: $o] :
( ( ord_less_eq_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A2: set_v,F: $o > set_v,B3: $o,C2: $o] :
( ( ord_less_eq_set_v @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_107_order__subst1,axiom,
! [A2: set_o,F: $o > set_o,B3: $o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ! [X4: $o,Y3: $o] :
( ( ord_less_eq_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A2: $o,F: set_v > $o,B3: set_v,C2: set_v] :
( ( ord_less_eq_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A2: $o,F: set_o > $o,B3: set_o,C2: set_o] :
( ( ord_less_eq_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: set_v,F: set_v > set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: set_v,F: set_o > set_v,B3: set_o,C2: set_o] :
( ( ord_less_eq_set_v @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A2: set_o,F: set_v > set_o,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ! [X4: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A2: set_o,F: set_o > set_o,B3: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ! [X4: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X4 @ Y3 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A2: $o,F: set_set_v > $o,B3: set_set_v,C2: set_set_v] :
( ( ord_less_eq_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ C2 )
=> ( ! [X4: set_set_v,Y3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X4 @ Y3 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_o @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_115_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z: set_v] : ( Y4 = Z ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_116_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_o,Z: set_o] : ( Y4 = Z ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
& ( ord_less_eq_set_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
= ( ^ [A4: $o,B4: $o] :
( ( ord_less_eq_o @ A4 @ B4 )
& ( ord_less_eq_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_v,Z: set_set_v] : ( Y4 = Z ) )
= ( ^ [A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B4 )
& ( ord_le5216385588623774835_set_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_se8455005133513928103od_v_v,Z: set_se8455005133513928103od_v_v] : ( Y4 = Z ) )
= ( ^ [A4: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A4 @ B4 )
& ( ord_le4714265922333009223od_v_v @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_antisym,axiom,
! [A2: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_122_antisym,axiom,
! [A2: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_123_antisym,axiom,
! [A2: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_124_antisym,axiom,
! [A2: $o,B3: $o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_less_eq_o @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [A2: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B3 )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [A2: set_se8455005133513928103od_v_v,B3: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B3 )
=> ( ( ord_le4714265922333009223od_v_v @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_127_dual__order_Otrans,axiom,
! [B3: set_v,A2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B3 @ A2 )
=> ( ( ord_less_eq_set_v @ C2 @ B3 )
=> ( ord_less_eq_set_v @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_128_dual__order_Otrans,axiom,
! [B3: set_Product_prod_v_v,A2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_129_dual__order_Otrans,axiom,
! [B3: set_o,A2: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( ( ord_less_eq_set_o @ C2 @ B3 )
=> ( ord_less_eq_set_o @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Otrans,axiom,
! [B3: $o,A2: $o,C2: $o] :
( ( ord_less_eq_o @ B3 @ A2 )
=> ( ( ord_less_eq_o @ C2 @ B3 )
=> ( ord_less_eq_o @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_131_dual__order_Otrans,axiom,
! [B3: set_set_v,A2: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B3 @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ C2 @ B3 )
=> ( ord_le5216385588623774835_set_v @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_132_dual__order_Otrans,axiom,
! [B3: set_se8455005133513928103od_v_v,A2: set_se8455005133513928103od_v_v,C2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ B3 @ A2 )
=> ( ( ord_le4714265922333009223od_v_v @ C2 @ B3 )
=> ( ord_le4714265922333009223od_v_v @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_133_dual__order_Oantisym,axiom,
! [B3: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B3 @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_134_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_135_dual__order_Oantisym,axiom,
! [B3: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_136_dual__order_Oantisym,axiom,
! [B3: $o,A2: $o] :
( ( ord_less_eq_o @ B3 @ A2 )
=> ( ( ord_less_eq_o @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_137_dual__order_Oantisym,axiom,
! [B3: set_set_v,A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B3 @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_138_dual__order_Oantisym,axiom,
! [B3: set_se8455005133513928103od_v_v,A2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ B3 @ A2 )
=> ( ( ord_le4714265922333009223od_v_v @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_139_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z: set_v] : ( Y4 = Z ) )
= ( ^ [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A4 )
& ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_140_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
& ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_141_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_o,Z: set_o] : ( Y4 = Z ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_142_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
= ( ^ [A4: $o,B4: $o] :
( ( ord_less_eq_o @ B4 @ A4 )
& ( ord_less_eq_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_143_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_v,Z: set_set_v] : ( Y4 = Z ) )
= ( ^ [A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B4 @ A4 )
& ( ord_le5216385588623774835_set_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_144_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_se8455005133513928103od_v_v,Z: set_se8455005133513928103od_v_v] : ( Y4 = Z ) )
= ( ^ [A4: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ B4 @ A4 )
& ( ord_le4714265922333009223od_v_v @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_145_linorder__wlog,axiom,
! [P: $o > $o > $o,A2: $o,B3: $o] :
( ! [A5: $o,B5: $o] :
( ( ord_less_eq_o @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: $o,B5: $o] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_146_order__trans,axiom,
! [X5: set_v,Y2: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X5 @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z2 )
=> ( ord_less_eq_set_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_147_order__trans,axiom,
! [X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_148_order__trans,axiom,
! [X5: set_o,Y2: set_o,Z2: set_o] :
( ( ord_less_eq_set_o @ X5 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ Z2 )
=> ( ord_less_eq_set_o @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_149_order__trans,axiom,
! [X5: $o,Y2: $o,Z2: $o] :
( ( ord_less_eq_o @ X5 @ Y2 )
=> ( ( ord_less_eq_o @ Y2 @ Z2 )
=> ( ord_less_eq_o @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_150_order__trans,axiom,
! [X5: set_set_v,Y2: set_set_v,Z2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ Y2 )
=> ( ( ord_le5216385588623774835_set_v @ Y2 @ Z2 )
=> ( ord_le5216385588623774835_set_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_151_order__trans,axiom,
! [X5: set_se8455005133513928103od_v_v,Y2: set_se8455005133513928103od_v_v,Z2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ X5 @ Y2 )
=> ( ( ord_le4714265922333009223od_v_v @ Y2 @ Z2 )
=> ( ord_le4714265922333009223od_v_v @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_152_order_Otrans,axiom,
! [A2: set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_153_order_Otrans,axiom,
! [A2: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_154_order_Otrans,axiom,
! [A2: set_o,B3: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ord_less_eq_set_o @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_155_order_Otrans,axiom,
! [A2: $o,B3: $o,C2: $o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ord_less_eq_o @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_156_order_Otrans,axiom,
! [A2: set_set_v,B3: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B3 )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ C2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_157_order_Otrans,axiom,
! [A2: set_se8455005133513928103od_v_v,B3: set_se8455005133513928103od_v_v,C2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B3 )
=> ( ( ord_le4714265922333009223od_v_v @ B3 @ C2 )
=> ( ord_le4714265922333009223od_v_v @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_158_order__antisym,axiom,
! [X5: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X5 @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_159_order__antisym,axiom,
! [X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_160_order__antisym,axiom,
! [X5: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X5 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_161_order__antisym,axiom,
! [X5: $o,Y2: $o] :
( ( ord_less_eq_o @ X5 @ Y2 )
=> ( ( ord_less_eq_o @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_162_order__antisym,axiom,
! [X5: set_set_v,Y2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X5 @ Y2 )
=> ( ( ord_le5216385588623774835_set_v @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_163_order__antisym,axiom,
! [X5: set_se8455005133513928103od_v_v,Y2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ X5 @ Y2 )
=> ( ( ord_le4714265922333009223od_v_v @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_164_ord__le__eq__trans,axiom,
! [A2: set_v,B3: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_165_ord__le__eq__trans,axiom,
! [A2: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_166_ord__le__eq__trans,axiom,
! [A2: set_o,B3: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_o @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_167_ord__le__eq__trans,axiom,
! [A2: $o,B3: $o,C2: $o] :
( ( ord_less_eq_o @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_o @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_168_ord__le__eq__trans,axiom,
! [A2: set_set_v,B3: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_169_ord__le__eq__trans,axiom,
! [A2: set_se8455005133513928103od_v_v,B3: set_se8455005133513928103od_v_v,C2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le4714265922333009223od_v_v @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_170_ord__eq__le__trans,axiom,
! [A2: set_v,B3: set_v,C2: set_v] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_171_ord__eq__le__trans,axiom,
! [A2: set_Product_prod_v_v,B3: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A2 = B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_172_ord__eq__le__trans,axiom,
! [A2: set_o,B3: set_o,C2: set_o] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_o @ B3 @ C2 )
=> ( ord_less_eq_set_o @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_173_ord__eq__le__trans,axiom,
! [A2: $o,B3: $o,C2: $o] :
( ( A2 = B3 )
=> ( ( ord_less_eq_o @ B3 @ C2 )
=> ( ord_less_eq_o @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_174_ord__eq__le__trans,axiom,
! [A2: set_set_v,B3: set_set_v,C2: set_set_v] :
( ( A2 = B3 )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ C2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_175_ord__eq__le__trans,axiom,
! [A2: set_se8455005133513928103od_v_v,B3: set_se8455005133513928103od_v_v,C2: set_se8455005133513928103od_v_v] :
( ( A2 = B3 )
=> ( ( ord_le4714265922333009223od_v_v @ B3 @ C2 )
=> ( ord_le4714265922333009223od_v_v @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_176_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z: set_v] : ( Y4 = Z ) )
= ( ^ [X2: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y )
& ( ord_less_eq_set_v @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_177_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y4 = Z ) )
= ( ^ [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y )
& ( ord_le7336532860387713383od_v_v @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_178_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_o,Z: set_o] : ( Y4 = Z ) )
= ( ^ [X2: set_o,Y: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y )
& ( ord_less_eq_set_o @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_179_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
= ( ^ [X2: $o,Y: $o] :
( ( ord_less_eq_o @ X2 @ Y )
& ( ord_less_eq_o @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_180_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_v,Z: set_set_v] : ( Y4 = Z ) )
= ( ^ [X2: set_set_v,Y: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X2 @ Y )
& ( ord_le5216385588623774835_set_v @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_181_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_se8455005133513928103od_v_v,Z: set_se8455005133513928103od_v_v] : ( Y4 = Z ) )
= ( ^ [X2: set_se8455005133513928103od_v_v,Y: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ X2 @ Y )
& ( ord_le4714265922333009223od_v_v @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_182_le__cases3,axiom,
! [X5: $o,Y2: $o,Z2: $o] :
( ( ( ord_less_eq_o @ X5 @ Y2 )
=> ~ ( ord_less_eq_o @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_o @ Y2 @ X5 )
=> ~ ( ord_less_eq_o @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_o @ X5 @ Z2 )
=> ~ ( ord_less_eq_o @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_o @ Z2 @ Y2 )
=> ~ ( ord_less_eq_o @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_o @ Y2 @ Z2 )
=> ~ ( ord_less_eq_o @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_o @ Z2 @ X5 )
=> ~ ( ord_less_eq_o @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_183_nle__le,axiom,
! [A2: $o,B3: $o] :
( ( ~ ( ord_less_eq_o @ A2 @ B3 ) )
= ( ( ord_less_eq_o @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_184_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_185_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_186_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_187_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_188_Collect__mono__iff,axiom,
! [P: set_Product_prod_v_v > $o,Q: set_Product_prod_v_v > $o] :
( ( ord_le4714265922333009223od_v_v @ ( collec8263177866097347122od_v_v @ P ) @ ( collec8263177866097347122od_v_v @ Q ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_189_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z: set_v] : ( Y4 = Z ) )
= ( ^ [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
& ( ord_less_eq_set_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_190_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y4 = Z ) )
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_191_set__eq__subset,axiom,
( ( ^ [Y4: set_o,Z: set_o] : ( Y4 = Z ) )
= ( ^ [A3: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
& ( ord_less_eq_set_o @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_192_set__eq__subset,axiom,
( ( ^ [Y4: set_set_v,Z: set_set_v] : ( Y4 = Z ) )
= ( ^ [A3: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_193_set__eq__subset,axiom,
( ( ^ [Y4: set_se8455005133513928103od_v_v,Z: set_se8455005133513928103od_v_v] : ( Y4 = Z ) )
= ( ^ [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B2 )
& ( ord_le4714265922333009223od_v_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_194_mem__Collect__eq,axiom,
! [A2: v,P: v > $o] :
( ( member_v @ A2 @ ( collect_v @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
! [A2: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
! [A2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( member8406446414694345712od_v_v @ A2 @ ( collec8263177866097347122od_v_v @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
! [A2: set_o,P: set_o > $o] :
( ( member_set_o @ A2 @ ( collect_set_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
! [A2: set_v,P: set_v > $o] :
( ( member_set_v @ A2 @ ( collect_set_v @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
! [A: set_set_o] :
( ( collect_set_o
@ ^ [X2: set_o] : ( member_set_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
! [A: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_206_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_207_subset__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_208_subset__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_209_subset__trans,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% subset_trans
thf(fact_210_subset__trans,axiom,
! [A: set_set_v,B: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( ord_le5216385588623774835_set_v @ B @ C )
=> ( ord_le5216385588623774835_set_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_211_subset__trans,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v,C: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( ord_le4714265922333009223od_v_v @ B @ C )
=> ( ord_le4714265922333009223od_v_v @ A @ C ) ) ) ).
% subset_trans
thf(fact_212_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X4: v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_213_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X4: product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_214_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X4: $o] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_215_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_216_Collect__mono,axiom,
! [P: set_Product_prod_v_v > $o,Q: set_Product_prod_v_v > $o] :
( ! [X4: set_Product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le4714265922333009223od_v_v @ ( collec8263177866097347122od_v_v @ P ) @ ( collec8263177866097347122od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_217_subset__refl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% subset_refl
thf(fact_218_subset__refl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% subset_refl
thf(fact_219_subset__refl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% subset_refl
thf(fact_220_subset__refl,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ A ) ).
% subset_refl
thf(fact_221_subset__refl,axiom,
! [A: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ A @ A ) ).
% subset_refl
thf(fact_222_subset__iff,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A3: set_set_o,B2: set_set_o] :
! [T: set_o] :
( ( member_set_o @ T @ A3 )
=> ( member_set_o @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_223_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
! [T: v] :
( ( member_v @ T @ A3 )
=> ( member_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_224_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A3 )
=> ( member7453568604450474000od_v_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_225_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_226_subset__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
! [T: set_v] :
( ( member_set_v @ T @ A3 )
=> ( member_set_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_227_subset__iff,axiom,
( ord_le4714265922333009223od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
! [T: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ T @ A3 )
=> ( member8406446414694345712od_v_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_228_equalityD2,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% equalityD2
thf(fact_229_equalityD2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% equalityD2
thf(fact_230_equalityD2,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ B @ A ) ) ).
% equalityD2
thf(fact_231_equalityD2,axiom,
! [A: set_set_v,B: set_set_v] :
( ( A = B )
=> ( ord_le5216385588623774835_set_v @ B @ A ) ) ).
% equalityD2
thf(fact_232_equalityD2,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( A = B )
=> ( ord_le4714265922333009223od_v_v @ B @ A ) ) ).
% equalityD2
thf(fact_233_equalityD1,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% equalityD1
thf(fact_234_equalityD1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% equalityD1
thf(fact_235_equalityD1,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% equalityD1
thf(fact_236_equalityD1,axiom,
! [A: set_set_v,B: set_set_v] :
( ( A = B )
=> ( ord_le5216385588623774835_set_v @ A @ B ) ) ).
% equalityD1
thf(fact_237_equalityD1,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( A = B )
=> ( ord_le4714265922333009223od_v_v @ A @ B ) ) ).
% equalityD1
thf(fact_238_subset__eq,axiom,
( ord_le4374716579403074808_set_o
= ( ^ [A3: set_set_o,B2: set_set_o] :
! [X2: set_o] :
( ( member_set_o @ X2 @ A3 )
=> ( member_set_o @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_239_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ( member_v @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_240_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_241_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_242_subset__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
! [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( member_set_v @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_243_subset__eq,axiom,
( ord_le4714265922333009223od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( member8406446414694345712od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_244_equalityE,axiom,
! [A: set_v,B: set_v] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ B @ A ) ) ) ).
% equalityE
thf(fact_245_equalityE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A ) ) ) ).
% equalityE
thf(fact_246_equalityE,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_o @ A @ B )
=> ~ ( ord_less_eq_set_o @ B @ A ) ) ) ).
% equalityE
thf(fact_247_equalityE,axiom,
! [A: set_set_v,B: set_set_v] :
( ( A = B )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ B )
=> ~ ( ord_le5216385588623774835_set_v @ B @ A ) ) ) ).
% equalityE
thf(fact_248_equalityE,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( A = B )
=> ~ ( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ~ ( ord_le4714265922333009223od_v_v @ B @ A ) ) ) ).
% equalityE
thf(fact_249_subsetD,axiom,
! [A: set_set_o,B: set_set_o,C2: set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ( member_set_o @ C2 @ A )
=> ( member_set_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_250_subsetD,axiom,
! [A: set_v,B: set_v,C2: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ C2 @ A )
=> ( member_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_251_subsetD,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_252_subsetD,axiom,
! [A: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_253_subsetD,axiom,
! [A: set_set_v,B: set_set_v,C2: set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( member_set_v @ C2 @ A )
=> ( member_set_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_254_subsetD,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v,C2: set_Product_prod_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( member8406446414694345712od_v_v @ C2 @ A )
=> ( member8406446414694345712od_v_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_255_in__mono,axiom,
! [A: set_set_o,B: set_set_o,X5: set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ( member_set_o @ X5 @ A )
=> ( member_set_o @ X5 @ B ) ) ) ).
% in_mono
thf(fact_256_in__mono,axiom,
! [A: set_v,B: set_v,X5: v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( member_v @ X5 @ A )
=> ( member_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_257_in__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X5: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( member7453568604450474000od_v_v @ X5 @ A )
=> ( member7453568604450474000od_v_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_258_in__mono,axiom,
! [A: set_o,B: set_o,X5: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X5 @ A )
=> ( member_o @ X5 @ B ) ) ) ).
% in_mono
thf(fact_259_in__mono,axiom,
! [A: set_set_v,B: set_set_v,X5: set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( member_set_v @ X5 @ A )
=> ( member_set_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_260_in__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( member8406446414694345712od_v_v @ X5 @ A )
=> ( member8406446414694345712od_v_v @ X5 @ B ) ) ) ).
% in_mono
thf(fact_261_UnionE,axiom,
! [A: set_v,C: set_set_set_v] :
( ( member_set_v @ A @ ( comple5450237519782573632_set_v @ C ) )
=> ~ ! [X6: set_set_v] :
( ( member_set_v @ A @ X6 )
=> ~ ( member_set_set_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_262_UnionE,axiom,
! [A: set_Product_prod_v_v,C: set_se2157405561750842759od_v_v] :
( ( member8406446414694345712od_v_v @ A @ ( comple6800410395631251988od_v_v @ C ) )
=> ~ ! [X6: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ A @ X6 )
=> ~ ( member5511408251247217616od_v_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_263_UnionE,axiom,
! [A: set_o,C: set_set_set_o] :
( ( member_set_o @ A @ ( comple4436988014476444997_set_o @ C ) )
=> ~ ! [X6: set_set_o] :
( ( member_set_o @ A @ X6 )
=> ~ ( member_set_set_o @ X6 @ C ) ) ) ).
% UnionE
thf(fact_264_UnionE,axiom,
! [A: v,C: set_set_v] :
( ( member_v @ A @ ( comple2307003700295860064_set_v @ C ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A @ X6 )
=> ~ ( member_set_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_265_UnionE,axiom,
! [A: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( comple5788137035815166516od_v_v @ C ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_266_UnionE,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A @ X6 )
=> ~ ( member_set_o @ X6 @ C ) ) ) ).
% UnionE
thf(fact_267_Collect__subset,axiom,
! [A: set_set_o,P: set_o > $o] :
( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_268_Collect__subset,axiom,
! [A: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_269_Collect__subset,axiom,
! [A: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_270_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_271_Collect__subset,axiom,
! [A: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_272_Collect__subset,axiom,
! [A: set_se8455005133513928103od_v_v,P: set_Product_prod_v_v > $o] :
( ord_le4714265922333009223od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_273_Sup__upper2,axiom,
! [U: set_set_v,A: set_set_set_v,V: set_set_v] :
( ( member_set_set_v @ U @ A )
=> ( ( ord_le5216385588623774835_set_v @ V @ U )
=> ( ord_le5216385588623774835_set_v @ V @ ( comple5450237519782573632_set_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_274_Sup__upper2,axiom,
! [U: set_se8455005133513928103od_v_v,A: set_se2157405561750842759od_v_v,V: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ U @ A )
=> ( ( ord_le4714265922333009223od_v_v @ V @ U )
=> ( ord_le4714265922333009223od_v_v @ V @ ( comple6800410395631251988od_v_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_275_Sup__upper2,axiom,
! [U: set_v,A: set_set_v,V: set_v] :
( ( member_set_v @ U @ A )
=> ( ( ord_less_eq_set_v @ V @ U )
=> ( ord_less_eq_set_v @ V @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_276_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_277_Sup__upper2,axiom,
! [U: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,V: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U @ A )
=> ( ( ord_le7336532860387713383od_v_v @ V @ U )
=> ( ord_le7336532860387713383od_v_v @ V @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% Sup_upper2
thf(fact_278_Sup__upper2,axiom,
! [U: set_o,A: set_set_o,V: set_o] :
( ( member_set_o @ U @ A )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_279_Sup__le__iff,axiom,
! [A: set_set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ B3 )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ A )
=> ( ord_le5216385588623774835_set_v @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_280_Sup__le__iff,axiom,
! [A: set_se2157405561750842759od_v_v,B3: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ B3 )
= ( ! [X2: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X2 @ A )
=> ( ord_le4714265922333009223od_v_v @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_281_Sup__le__iff,axiom,
! [A: set_set_v,B3: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ B3 )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( ord_less_eq_set_v @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_282_Sup__le__iff,axiom,
! [A: set_o,B3: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B3 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_283_Sup__le__iff,axiom,
! [A: set_se8455005133513928103od_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ B3 )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( ord_le7336532860387713383od_v_v @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_284_Sup__le__iff,axiom,
! [A: set_set_o,B3: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ B3 )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( ord_less_eq_set_o @ X2 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_285_Sup__upper,axiom,
! [X5: set_set_v,A: set_set_set_v] :
( ( member_set_set_v @ X5 @ A )
=> ( ord_le5216385588623774835_set_v @ X5 @ ( comple5450237519782573632_set_v @ A ) ) ) ).
% Sup_upper
thf(fact_286_Sup__upper,axiom,
! [X5: set_se8455005133513928103od_v_v,A: set_se2157405561750842759od_v_v] :
( ( member5511408251247217616od_v_v @ X5 @ A )
=> ( ord_le4714265922333009223od_v_v @ X5 @ ( comple6800410395631251988od_v_v @ A ) ) ) ).
% Sup_upper
thf(fact_287_Sup__upper,axiom,
! [X5: set_v,A: set_set_v] :
( ( member_set_v @ X5 @ A )
=> ( ord_less_eq_set_v @ X5 @ ( comple2307003700295860064_set_v @ A ) ) ) ).
% Sup_upper
thf(fact_288_Sup__upper,axiom,
! [X5: $o,A: set_o] :
( ( member_o @ X5 @ A )
=> ( ord_less_eq_o @ X5 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_289_Sup__upper,axiom,
! [X5: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ A )
=> ( ord_le7336532860387713383od_v_v @ X5 @ ( comple5788137035815166516od_v_v @ A ) ) ) ).
% Sup_upper
thf(fact_290_Sup__upper,axiom,
! [X5: set_o,A: set_set_o] :
( ( member_set_o @ X5 @ A )
=> ( ord_less_eq_set_o @ X5 @ ( comple90263536869209701_set_o @ A ) ) ) ).
% Sup_upper
thf(fact_291_Sup__least,axiom,
! [A: set_set_set_v,Z2: set_set_v] :
( ! [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ X4 @ Z2 ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_292_Sup__least,axiom,
! [A: set_se2157405561750842759od_v_v,Z2: set_se8455005133513928103od_v_v] :
( ! [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
=> ( ord_le4714265922333009223od_v_v @ X4 @ Z2 ) )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_293_Sup__least,axiom,
! [A: set_set_v,Z2: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_v @ X4 @ Z2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_294_Sup__least,axiom,
! [A: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_295_Sup__least,axiom,
! [A: set_se8455005133513928103od_v_v,Z2: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_296_Sup__least,axiom,
! [A: set_set_o,Z2: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_set_o @ X4 @ Z2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_297_Sup__mono,axiom,
! [A: set_set_set_v,B: set_set_set_v] :
( ! [A5: set_set_v] :
( ( member_set_set_v @ A5 @ A )
=> ? [X: set_set_v] :
( ( member_set_set_v @ X @ B )
& ( ord_le5216385588623774835_set_v @ A5 @ X ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Sup_mono
thf(fact_298_Sup__mono,axiom,
! [A: set_se2157405561750842759od_v_v,B: set_se2157405561750842759od_v_v] :
( ! [A5: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ A5 @ A )
=> ? [X: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X @ B )
& ( ord_le4714265922333009223od_v_v @ A5 @ X ) ) )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ ( comple6800410395631251988od_v_v @ B ) ) ) ).
% Sup_mono
thf(fact_299_Sup__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [A5: set_v] :
( ( member_set_v @ A5 @ A )
=> ? [X: set_v] :
( ( member_set_v @ X @ B )
& ( ord_less_eq_set_v @ A5 @ X ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_mono
thf(fact_300_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B )
& ( ord_less_eq_o @ A5 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_301_Sup__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [A5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A5 @ A )
=> ? [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A5 @ X ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_mono
thf(fact_302_Sup__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A )
=> ? [X: set_o] :
( ( member_set_o @ X @ B )
& ( ord_less_eq_set_o @ A5 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Sup_mono
thf(fact_303_Sup__eqI,axiom,
! [A: set_set_set_v,X5: set_set_v] :
( ! [Y3: set_set_v] :
( ( member_set_set_v @ Y3 @ A )
=> ( ord_le5216385588623774835_set_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_set_v] :
( ! [Z3: set_set_v] :
( ( member_set_set_v @ Z3 @ A )
=> ( ord_le5216385588623774835_set_v @ Z3 @ Y3 ) )
=> ( ord_le5216385588623774835_set_v @ X5 @ Y3 ) )
=> ( ( comple5450237519782573632_set_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_304_Sup__eqI,axiom,
! [A: set_se2157405561750842759od_v_v,X5: set_se8455005133513928103od_v_v] :
( ! [Y3: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Y3 @ A )
=> ( ord_le4714265922333009223od_v_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_se8455005133513928103od_v_v] :
( ! [Z3: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Z3 @ A )
=> ( ord_le4714265922333009223od_v_v @ Z3 @ Y3 ) )
=> ( ord_le4714265922333009223od_v_v @ X5 @ Y3 ) )
=> ( ( comple6800410395631251988od_v_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_305_Sup__eqI,axiom,
! [A: set_set_v,X5: set_v] :
( ! [Y3: set_v] :
( ( member_set_v @ Y3 @ A )
=> ( ord_less_eq_set_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_v] :
( ! [Z3: set_v] :
( ( member_set_v @ Z3 @ A )
=> ( ord_less_eq_set_v @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_v @ X5 @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_306_Sup__eqI,axiom,
! [A: set_o,X5: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ! [Y3: $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ A )
=> ( ord_less_eq_o @ Z3 @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_307_Sup__eqI,axiom,
! [A: set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v] :
( ! [Y3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y3 @ A )
=> ( ord_le7336532860387713383od_v_v @ Y3 @ X5 ) )
=> ( ! [Y3: set_Product_prod_v_v] :
( ! [Z3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z3 @ A )
=> ( ord_le7336532860387713383od_v_v @ Z3 @ Y3 ) )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Y3 ) )
=> ( ( comple5788137035815166516od_v_v @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_308_Sup__eqI,axiom,
! [A: set_set_o,X5: set_o] :
( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A )
=> ( ord_less_eq_set_o @ Y3 @ X5 ) )
=> ( ! [Y3: set_o] :
( ! [Z3: set_o] :
( ( member_set_o @ Z3 @ A )
=> ( ord_less_eq_set_o @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_o @ X5 @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_309_Union__subsetI,axiom,
! [A: set_set_set_v,B: set_set_set_v] :
( ! [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
=> ? [Y5: set_set_v] :
( ( member_set_set_v @ Y5 @ B )
& ( ord_le5216385588623774835_set_v @ X4 @ Y5 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Union_subsetI
thf(fact_310_Union__subsetI,axiom,
! [A: set_se2157405561750842759od_v_v,B: set_se2157405561750842759od_v_v] :
( ! [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
=> ? [Y5: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Y5 @ B )
& ( ord_le4714265922333009223od_v_v @ X4 @ Y5 ) ) )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ ( comple6800410395631251988od_v_v @ B ) ) ) ).
% Union_subsetI
thf(fact_311_Union__subsetI,axiom,
! [A: set_set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ? [Y5: set_v] :
( ( member_set_v @ Y5 @ B )
& ( ord_less_eq_set_v @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_subsetI
thf(fact_312_Union__subsetI,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ? [Y5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y5 @ B )
& ( ord_le7336532860387713383od_v_v @ X4 @ Y5 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_subsetI
thf(fact_313_Union__subsetI,axiom,
! [A: set_set_o,B: set_set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ? [Y5: set_o] :
( ( member_set_o @ Y5 @ B )
& ( ord_less_eq_set_o @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Union_subsetI
thf(fact_314_Union__upper,axiom,
! [B: set_set_v,A: set_set_set_v] :
( ( member_set_set_v @ B @ A )
=> ( ord_le5216385588623774835_set_v @ B @ ( comple5450237519782573632_set_v @ A ) ) ) ).
% Union_upper
thf(fact_315_Union__upper,axiom,
! [B: set_se8455005133513928103od_v_v,A: set_se2157405561750842759od_v_v] :
( ( member5511408251247217616od_v_v @ B @ A )
=> ( ord_le4714265922333009223od_v_v @ B @ ( comple6800410395631251988od_v_v @ A ) ) ) ).
% Union_upper
thf(fact_316_Union__upper,axiom,
! [B: set_v,A: set_set_v] :
( ( member_set_v @ B @ A )
=> ( ord_less_eq_set_v @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ).
% Union_upper
thf(fact_317_Union__upper,axiom,
! [B: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B @ A )
=> ( ord_le7336532860387713383od_v_v @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ).
% Union_upper
thf(fact_318_Union__upper,axiom,
! [B: set_o,A: set_set_o] :
( ( member_set_o @ B @ A )
=> ( ord_less_eq_set_o @ B @ ( comple90263536869209701_set_o @ A ) ) ) ).
% Union_upper
thf(fact_319_Union__least,axiom,
! [A: set_set_set_v,C: set_set_v] :
( ! [X6: set_set_v] :
( ( member_set_set_v @ X6 @ A )
=> ( ord_le5216385588623774835_set_v @ X6 @ C ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ A ) @ C ) ) ).
% Union_least
thf(fact_320_Union__least,axiom,
! [A: set_se2157405561750842759od_v_v,C: set_se8455005133513928103od_v_v] :
( ! [X6: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X6 @ A )
=> ( ord_le4714265922333009223od_v_v @ X6 @ C ) )
=> ( ord_le4714265922333009223od_v_v @ ( comple6800410395631251988od_v_v @ A ) @ C ) ) ).
% Union_least
thf(fact_321_Union__least,axiom,
! [A: set_set_v,C: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A )
=> ( ord_less_eq_set_v @ X6 @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A ) @ C ) ) ).
% Union_least
thf(fact_322_Union__least,axiom,
! [A: set_se8455005133513928103od_v_v,C: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A ) @ C ) ) ).
% Union_least
thf(fact_323_Union__least,axiom,
! [A: set_set_o,C: set_o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ A )
=> ( ord_less_eq_set_o @ X6 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ C ) ) ).
% Union_least
thf(fact_324_cSup__eq__maximum,axiom,
! [Z2: set_set_v,X3: set_set_set_v] :
( ( member_set_set_v @ Z2 @ X3 )
=> ( ! [X4: set_set_v] :
( ( member_set_set_v @ X4 @ X3 )
=> ( ord_le5216385588623774835_set_v @ X4 @ Z2 ) )
=> ( ( comple5450237519782573632_set_v @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_325_cSup__eq__maximum,axiom,
! [Z2: set_se8455005133513928103od_v_v,X3: set_se2157405561750842759od_v_v] :
( ( member5511408251247217616od_v_v @ Z2 @ X3 )
=> ( ! [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ X3 )
=> ( ord_le4714265922333009223od_v_v @ X4 @ Z2 ) )
=> ( ( comple6800410395631251988od_v_v @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_326_cSup__eq__maximum,axiom,
! [Z2: set_v,X3: set_set_v] :
( ( member_set_v @ Z2 @ X3 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ X3 )
=> ( ord_less_eq_set_v @ X4 @ Z2 ) )
=> ( ( comple2307003700295860064_set_v @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_327_cSup__eq__maximum,axiom,
! [Z2: $o,X3: set_o] :
( ( member_o @ Z2 @ X3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X3 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_o @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_328_cSup__eq__maximum,axiom,
! [Z2: set_Product_prod_v_v,X3: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ Z2 @ X3 )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ Z2 ) )
=> ( ( comple5788137035815166516od_v_v @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_329_cSup__eq__maximum,axiom,
! [Z2: set_o,X3: set_set_o] :
( ( member_set_o @ Z2 @ X3 )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ X3 )
=> ( ord_less_eq_set_o @ X4 @ Z2 ) )
=> ( ( comple90263536869209701_set_o @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_330_w_I4_J,axiom,
~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ e ) ) ).
% w(4)
thf(fact_331_w_I3_J,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% w(3)
thf(fact_332_e_H__def,axiom,
( e2
= ( sCC_Bloemen_unite_v @ v2 @ w @ e ) ) ).
% e'_def
thf(fact_333_pred__subset__eq,axiom,
! [R: set_set_o,S: set_set_o] :
( ( ord_less_eq_set_o_o
@ ^ [X2: set_o] : ( member_set_o @ X2 @ R )
@ ^ [X2: set_o] : ( member_set_o @ X2 @ S ) )
= ( ord_le4374716579403074808_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_334_pred__subset__eq,axiom,
! [R: set_v,S: set_v] :
( ( ord_less_eq_v_o
@ ^ [X2: v] : ( member_v @ X2 @ R )
@ ^ [X2: v] : ( member_v @ X2 @ S ) )
= ( ord_less_eq_set_v @ R @ S ) ) ).
% pred_subset_eq
thf(fact_335_pred__subset__eq,axiom,
! [R: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( ord_le5892402249245633078_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ R )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ S ) )
= ( ord_le7336532860387713383od_v_v @ R @ S ) ) ).
% pred_subset_eq
thf(fact_336_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_337_pred__subset__eq,axiom,
! [R: set_set_v,S: set_set_v] :
( ( ord_less_eq_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ R )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ S ) )
= ( ord_le5216385588623774835_set_v @ R @ S ) ) ).
% pred_subset_eq
thf(fact_338_pred__subset__eq,axiom,
! [R: set_se8455005133513928103od_v_v,S: set_se8455005133513928103od_v_v] :
( ( ord_le2647354080893881942_v_v_o
@ ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ R )
@ ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ S ) )
= ( ord_le4714265922333009223od_v_v @ R @ S ) ) ).
% pred_subset_eq
thf(fact_339_subset__Collect__iff,axiom,
! [B: set_set_o,A: set_set_o,P: set_o > $o] :
( ( ord_le4374716579403074808_set_o @ B @ A )
=> ( ( ord_le4374716579403074808_set_o @ B
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_340_subset__Collect__iff,axiom,
! [B: set_v,A: set_v,P: v > $o] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ B
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_341_subset__Collect__iff,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ B
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_342_subset__Collect__iff,axiom,
! [B: set_o,A: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_343_subset__Collect__iff,axiom,
! [B: set_set_v,A: set_set_v,P: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( ord_le5216385588623774835_set_v @ B
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_344_subset__Collect__iff,axiom,
! [B: set_se8455005133513928103od_v_v,A: set_se8455005133513928103od_v_v,P: set_Product_prod_v_v > $o] :
( ( ord_le4714265922333009223od_v_v @ B @ A )
=> ( ( ord_le4714265922333009223od_v_v @ B
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_345_subset__CollectI,axiom,
! [B: set_set_o,A: set_set_o,Q: set_o > $o,P: set_o > $o] :
( ( ord_le4374716579403074808_set_o @ B @ A )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_346_subset__CollectI,axiom,
! [B: set_v,A: set_v,Q: v > $o,P: v > $o] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ! [X4: v] :
( ( member_v @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_347_subset__CollectI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,Q: product_prod_v_v > $o,P: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_348_subset__CollectI,axiom,
! [B: set_o,A: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_349_subset__CollectI,axiom,
! [B: set_set_v,A: set_set_v,Q: set_v > $o,P: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_350_subset__CollectI,axiom,
! [B: set_se8455005133513928103od_v_v,A: set_se8455005133513928103od_v_v,Q: set_Product_prod_v_v > $o,P: set_Product_prod_v_v > $o] :
( ( ord_le4714265922333009223od_v_v @ B @ A )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ B )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_le4714265922333009223od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_351_Collect__restrict,axiom,
! [X3: set_set_o,P: set_o > $o] :
( ord_le4374716579403074808_set_o
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_352_Collect__restrict,axiom,
! [X3: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_353_Collect__restrict,axiom,
! [X3: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_354_Collect__restrict,axiom,
! [X3: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_355_Collect__restrict,axiom,
! [X3: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_356_Collect__restrict,axiom,
! [X3: set_se8455005133513928103od_v_v,P: set_Product_prod_v_v > $o] :
( ord_le4714265922333009223od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_357_prop__restrict,axiom,
! [X5: set_o,Z4: set_set_o,X3: set_set_o,P: set_o > $o] :
( ( member_set_o @ X5 @ Z4 )
=> ( ( ord_le4374716579403074808_set_o @ Z4
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_358_prop__restrict,axiom,
! [X5: v,Z4: set_v,X3: set_v,P: v > $o] :
( ( member_v @ X5 @ Z4 )
=> ( ( ord_less_eq_set_v @ Z4
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_359_prop__restrict,axiom,
! [X5: product_prod_v_v,Z4: set_Product_prod_v_v,X3: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X5 @ Z4 )
=> ( ( ord_le7336532860387713383od_v_v @ Z4
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_360_prop__restrict,axiom,
! [X5: $o,Z4: set_o,X3: set_o,P: $o > $o] :
( ( member_o @ X5 @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_361_prop__restrict,axiom,
! [X5: set_v,Z4: set_set_v,X3: set_set_v,P: set_v > $o] :
( ( member_set_v @ X5 @ Z4 )
=> ( ( ord_le5216385588623774835_set_v @ Z4
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_362_prop__restrict,axiom,
! [X5: set_Product_prod_v_v,Z4: set_se8455005133513928103od_v_v,X3: set_se8455005133513928103od_v_v,P: set_Product_prod_v_v > $o] :
( ( member8406446414694345712od_v_v @ X5 @ Z4 )
=> ( ( ord_le4714265922333009223od_v_v @ Z4
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_363_conj__subset__def,axiom,
! [A: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A
@ ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_v @ A @ ( collect_v @ P ) )
& ( ord_less_eq_set_v @ A @ ( collect_v @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_364_conj__subset__def,axiom,
! [A: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
& ( ord_le7336532860387713383od_v_v @ A @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_365_conj__subset__def,axiom,
! [A: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A
@ ( collect_o
@ ^ [X2: $o] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_o @ A @ ( collect_o @ P ) )
& ( ord_less_eq_set_o @ A @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_366_conj__subset__def,axiom,
! [A: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le5216385588623774835_set_v @ A @ ( collect_set_v @ P ) )
& ( ord_le5216385588623774835_set_v @ A @ ( collect_set_v @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_367_conj__subset__def,axiom,
! [A: set_se8455005133513928103od_v_v,P: set_Product_prod_v_v > $o,Q: set_Product_prod_v_v > $o] :
( ( ord_le4714265922333009223od_v_v @ A
@ ( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le4714265922333009223od_v_v @ A @ ( collec8263177866097347122od_v_v @ P ) )
& ( ord_le4714265922333009223od_v_v @ A @ ( collec8263177866097347122od_v_v @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_368_Greatest__equality,axiom,
! [P: set_v > $o,X5: set_v] :
( ( P @ X5 )
=> ( ! [Y3: set_v] :
( ( P @ Y3 )
=> ( ord_less_eq_set_v @ Y3 @ X5 ) )
=> ( ( order_Greatest_set_v @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_369_Greatest__equality,axiom,
! [P: set_Product_prod_v_v > $o,X5: set_Product_prod_v_v] :
( ( P @ X5 )
=> ( ! [Y3: set_Product_prod_v_v] :
( ( P @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ Y3 @ X5 ) )
=> ( ( order_3682441541413997216od_v_v @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_370_Greatest__equality,axiom,
! [P: set_o > $o,X5: set_o] :
( ( P @ X5 )
=> ( ! [Y3: set_o] :
( ( P @ Y3 )
=> ( ord_less_eq_set_o @ Y3 @ X5 ) )
=> ( ( order_Greatest_set_o @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_371_Greatest__equality,axiom,
! [P: $o > $o,X5: $o] :
( ( P @ X5 )
=> ( ! [Y3: $o] :
( ( P @ Y3 )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( order_Greatest_o @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_372_Greatest__equality,axiom,
! [P: set_set_v > $o,X5: set_set_v] :
( ( P @ X5 )
=> ( ! [Y3: set_set_v] :
( ( P @ Y3 )
=> ( ord_le5216385588623774835_set_v @ Y3 @ X5 ) )
=> ( ( order_5057575371121949356_set_v @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_373_Greatest__equality,axiom,
! [P: set_se8455005133513928103od_v_v > $o,X5: set_se8455005133513928103od_v_v] :
( ( P @ X5 )
=> ( ! [Y3: set_se8455005133513928103od_v_v] :
( ( P @ Y3 )
=> ( ord_le4714265922333009223od_v_v @ Y3 @ X5 ) )
=> ( ( order_447341434510879360od_v_v @ P )
= X5 ) ) ) ).
% Greatest_equality
thf(fact_374_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl3795065053823578884t_unit @ e @ v2 ) ) ).
% w(2)
thf(fact_375_GreatestI2__order,axiom,
! [P: set_v > $o,X5: set_v,Q: set_v > $o] :
( ( P @ X5 )
=> ( ! [Y3: set_v] :
( ( P @ Y3 )
=> ( ord_less_eq_set_v @ Y3 @ X5 ) )
=> ( ! [X4: set_v] :
( ( P @ X4 )
=> ( ! [Y5: set_v] :
( ( P @ Y5 )
=> ( ord_less_eq_set_v @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_set_v @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_376_GreatestI2__order,axiom,
! [P: set_Product_prod_v_v > $o,X5: set_Product_prod_v_v,Q: set_Product_prod_v_v > $o] :
( ( P @ X5 )
=> ( ! [Y3: set_Product_prod_v_v] :
( ( P @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ Y3 @ X5 ) )
=> ( ! [X4: set_Product_prod_v_v] :
( ( P @ X4 )
=> ( ! [Y5: set_Product_prod_v_v] :
( ( P @ Y5 )
=> ( ord_le7336532860387713383od_v_v @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_3682441541413997216od_v_v @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_377_GreatestI2__order,axiom,
! [P: set_o > $o,X5: set_o,Q: set_o > $o] :
( ( P @ X5 )
=> ( ! [Y3: set_o] :
( ( P @ Y3 )
=> ( ord_less_eq_set_o @ Y3 @ X5 ) )
=> ( ! [X4: set_o] :
( ( P @ X4 )
=> ( ! [Y5: set_o] :
( ( P @ Y5 )
=> ( ord_less_eq_set_o @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_set_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_378_GreatestI2__order,axiom,
! [P: $o > $o,X5: $o,Q: $o > $o] :
( ( P @ X5 )
=> ( ! [Y3: $o] :
( ( P @ Y3 )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ! [X4: $o] :
( ( P @ X4 )
=> ( ! [Y5: $o] :
( ( P @ Y5 )
=> ( ord_less_eq_o @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_379_GreatestI2__order,axiom,
! [P: set_set_v > $o,X5: set_set_v,Q: set_set_v > $o] :
( ( P @ X5 )
=> ( ! [Y3: set_set_v] :
( ( P @ Y3 )
=> ( ord_le5216385588623774835_set_v @ Y3 @ X5 ) )
=> ( ! [X4: set_set_v] :
( ( P @ X4 )
=> ( ! [Y5: set_set_v] :
( ( P @ Y5 )
=> ( ord_le5216385588623774835_set_v @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_5057575371121949356_set_v @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_380_GreatestI2__order,axiom,
! [P: set_se8455005133513928103od_v_v > $o,X5: set_se8455005133513928103od_v_v,Q: set_se8455005133513928103od_v_v > $o] :
( ( P @ X5 )
=> ( ! [Y3: set_se8455005133513928103od_v_v] :
( ( P @ Y3 )
=> ( ord_le4714265922333009223od_v_v @ Y3 @ X5 ) )
=> ( ! [X4: set_se8455005133513928103od_v_v] :
( ( P @ X4 )
=> ( ! [Y5: set_se8455005133513928103od_v_v] :
( ( P @ Y5 )
=> ( ord_le4714265922333009223od_v_v @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_447341434510879360od_v_v @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_381_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_382_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_383_SUP__subset__mono,axiom,
! [A: set_v,B: set_v,F: v > $o,G: v > $o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_v_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_384_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_385_SUP__subset__mono,axiom,
! [A: set_v,B: set_v,F: v > set_v,G: v > set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_386_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_v,G: $o > set_v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_387_SUP__subset__mono,axiom,
! [A: set_set_o,B: set_set_o,F: set_o > $o,G: set_o > $o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_388_SUP__subset__mono,axiom,
! [A: set_set_v,B: set_set_v,F: set_v > $o,G: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_v_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_v_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_389_SUP__subset__mono,axiom,
! [A: set_v,B: set_v,F: v > set_o,G: v > set_o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_v_set_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_390_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_391_SUP__subset__mono,axiom,
! [A: set_v,B: set_v,F: v > set_set_v,G: v > set_set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) ) @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_392_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_set_v,G: $o > set_set_v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) ) @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_393_unfold__congs_I3_J,axiom,
! [R2: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit,V2: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R2 = R3 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R3 )
= V2 )
=> ( ! [V3: set_v] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R2 )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R3 ) ) ) ) ) ).
% unfold_congs(3)
thf(fact_394_fold__congs_I3_J,axiom,
! [R2: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit,V2: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R2 = R3 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R3 )
= V2 )
=> ( ! [V3: set_v] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R2 )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R3 ) ) ) ) ) ).
% fold_congs(3)
thf(fact_395_verit__comp__simplify1_I2_J,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_396_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_397_verit__comp__simplify1_I2_J,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_398_verit__comp__simplify1_I2_J,axiom,
! [A2: $o] : ( ord_less_eq_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_399_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_400_verit__comp__simplify1_I2_J,axiom,
! [A2: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_401_verit__la__disequality,axiom,
! [A2: $o,B3: $o] :
( ( A2 = B3 )
| ~ ( ord_less_eq_o @ A2 @ B3 )
| ~ ( ord_less_eq_o @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_402_image__eqI,axiom,
! [B3: v,F: v > v,X5: v,A: set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_v @ X5 @ A )
=> ( member_v @ B3 @ ( image_v_v @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_403_image__eqI,axiom,
! [B3: $o,F: v > $o,X5: v,A: set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_v @ X5 @ A )
=> ( member_o @ B3 @ ( image_v_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_404_image__eqI,axiom,
! [B3: v,F: $o > v,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_v @ B3 @ ( image_o_v @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_405_image__eqI,axiom,
! [B3: $o,F: $o > $o,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_406_image__eqI,axiom,
! [B3: set_v,F: v > set_v,X5: v,A: set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_v @ X5 @ A )
=> ( member_set_v @ B3 @ ( image_v_set_v @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_407_image__eqI,axiom,
! [B3: set_o,F: v > set_o,X5: v,A: set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_v @ X5 @ A )
=> ( member_set_o @ B3 @ ( image_v_set_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_408_image__eqI,axiom,
! [B3: set_v,F: $o > set_v,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_set_v @ B3 @ ( image_o_set_v @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_409_image__eqI,axiom,
! [B3: set_o,F: $o > set_o,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_set_o @ B3 @ ( image_o_set_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_410_image__eqI,axiom,
! [B3: v,F: set_v > v,X5: set_v,A: set_set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_set_v @ X5 @ A )
=> ( member_v @ B3 @ ( image_set_v_v @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_411_image__eqI,axiom,
! [B3: $o,F: set_v > $o,X5: set_v,A: set_set_v] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_set_v @ X5 @ A )
=> ( member_o @ B3 @ ( image_set_v_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_412_UN__I,axiom,
! [A2: v,A: set_v,B3: v,B: v > set_v] :
( ( member_v @ A2 @ A )
=> ( ( member_v @ B3 @ ( B @ A2 ) )
=> ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_413_UN__I,axiom,
! [A2: $o,A: set_o,B3: v,B: $o > set_v] :
( ( member_o @ A2 @ A )
=> ( ( member_v @ B3 @ ( B @ A2 ) )
=> ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_414_UN__I,axiom,
! [A2: v,A: set_v,B3: $o,B: v > set_o] :
( ( member_v @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_415_UN__I,axiom,
! [A2: $o,A: set_o,B3: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_416_UN__I,axiom,
! [A2: v,A: set_v,B3: set_v,B: v > set_set_v] :
( ( member_v @ A2 @ A )
=> ( ( member_set_v @ B3 @ ( B @ A2 ) )
=> ( member_set_v @ B3 @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_417_UN__I,axiom,
! [A2: v,A: set_v,B3: set_o,B: v > set_set_o] :
( ( member_v @ A2 @ A )
=> ( ( member_set_o @ B3 @ ( B @ A2 ) )
=> ( member_set_o @ B3 @ ( comple4436988014476444997_set_o @ ( image_v_set_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_418_UN__I,axiom,
! [A2: $o,A: set_o,B3: set_v,B: $o > set_set_v] :
( ( member_o @ A2 @ A )
=> ( ( member_set_v @ B3 @ ( B @ A2 ) )
=> ( member_set_v @ B3 @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_419_UN__I,axiom,
! [A2: $o,A: set_o,B3: set_o,B: $o > set_set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_set_o @ B3 @ ( B @ A2 ) )
=> ( member_set_o @ B3 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_420_UN__I,axiom,
! [A2: set_v,A: set_set_v,B3: v,B: set_v > set_v] :
( ( member_set_v @ A2 @ A )
=> ( ( member_v @ B3 @ ( B @ A2 ) )
=> ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_421_UN__I,axiom,
! [A2: set_o,A: set_set_o,B3: v,B: set_o > set_v] :
( ( member_set_o @ A2 @ A )
=> ( ( member_v @ B3 @ ( B @ A2 ) )
=> ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_422_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_423_SUP__identity__eq,axiom,
! [A: set_set_v] :
( ( comple2307003700295860064_set_v
@ ( image_set_v_set_v
@ ^ [X2: set_v] : X2
@ A ) )
= ( comple2307003700295860064_set_v @ A ) ) ).
% SUP_identity_eq
thf(fact_424_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_425_SUP__identity__eq,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v
@ ( image_5212826947168092101od_v_v
@ ^ [X2: set_Product_prod_v_v] : X2
@ A ) )
= ( comple5788137035815166516od_v_v @ A ) ) ).
% SUP_identity_eq
thf(fact_426_SUP__identity__eq,axiom,
! [A: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X2: set_o] : X2
@ A ) )
= ( comple90263536869209701_set_o @ A ) ) ).
% SUP_identity_eq
thf(fact_427_sclosed,axiom,
! [X: v] :
( ( member_v @ X @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X ) @ vertices ) ) ).
% sclosed
thf(fact_428_rev__image__eqI,axiom,
! [X5: v,A: set_v,B3: v,F: v > v] :
( ( member_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_v @ B3 @ ( image_v_v @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_429_rev__image__eqI,axiom,
! [X5: v,A: set_v,B3: $o,F: v > $o] :
( ( member_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_v_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_430_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: v,F: $o > v] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_v @ B3 @ ( image_o_v @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_431_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: $o,F: $o > $o] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_432_rev__image__eqI,axiom,
! [X5: v,A: set_v,B3: set_v,F: v > set_v] :
( ( member_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_set_v @ B3 @ ( image_v_set_v @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_433_rev__image__eqI,axiom,
! [X5: v,A: set_v,B3: set_o,F: v > set_o] :
( ( member_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_set_o @ B3 @ ( image_v_set_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_434_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: set_v,F: $o > set_v] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_set_v @ B3 @ ( image_o_set_v @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_435_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: set_o,F: $o > set_o] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_set_o @ B3 @ ( image_o_set_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_436_rev__image__eqI,axiom,
! [X5: set_v,A: set_set_v,B3: v,F: set_v > v] :
( ( member_set_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_v @ B3 @ ( image_set_v_v @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_437_rev__image__eqI,axiom,
! [X5: set_v,A: set_set_v,B3: $o,F: set_v > $o] :
( ( member_set_v @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_set_v_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_438_imageI,axiom,
! [X5: v,A: set_v,F: v > v] :
( ( member_v @ X5 @ A )
=> ( member_v @ ( F @ X5 ) @ ( image_v_v @ F @ A ) ) ) ).
% imageI
thf(fact_439_imageI,axiom,
! [X5: v,A: set_v,F: v > $o] :
( ( member_v @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_v_o @ F @ A ) ) ) ).
% imageI
thf(fact_440_imageI,axiom,
! [X5: $o,A: set_o,F: $o > v] :
( ( member_o @ X5 @ A )
=> ( member_v @ ( F @ X5 ) @ ( image_o_v @ F @ A ) ) ) ).
% imageI
thf(fact_441_imageI,axiom,
! [X5: $o,A: set_o,F: $o > $o] :
( ( member_o @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_442_imageI,axiom,
! [X5: v,A: set_v,F: v > set_v] :
( ( member_v @ X5 @ A )
=> ( member_set_v @ ( F @ X5 ) @ ( image_v_set_v @ F @ A ) ) ) ).
% imageI
thf(fact_443_imageI,axiom,
! [X5: v,A: set_v,F: v > set_o] :
( ( member_v @ X5 @ A )
=> ( member_set_o @ ( F @ X5 ) @ ( image_v_set_o @ F @ A ) ) ) ).
% imageI
thf(fact_444_imageI,axiom,
! [X5: $o,A: set_o,F: $o > set_v] :
( ( member_o @ X5 @ A )
=> ( member_set_v @ ( F @ X5 ) @ ( image_o_set_v @ F @ A ) ) ) ).
% imageI
thf(fact_445_imageI,axiom,
! [X5: $o,A: set_o,F: $o > set_o] :
( ( member_o @ X5 @ A )
=> ( member_set_o @ ( F @ X5 ) @ ( image_o_set_o @ F @ A ) ) ) ).
% imageI
thf(fact_446_imageI,axiom,
! [X5: set_v,A: set_set_v,F: set_v > v] :
( ( member_set_v @ X5 @ A )
=> ( member_v @ ( F @ X5 ) @ ( image_set_v_v @ F @ A ) ) ) ).
% imageI
thf(fact_447_imageI,axiom,
! [X5: set_v,A: set_set_v,F: set_v > $o] :
( ( member_set_v @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_set_v_o @ F @ A ) ) ) ).
% imageI
thf(fact_448_imageE,axiom,
! [B3: v,F: v > v,A: set_v] :
( ( member_v @ B3 @ ( image_v_v @ F @ A ) )
=> ~ ! [X4: v] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_v @ X4 @ A ) ) ) ).
% imageE
thf(fact_449_imageE,axiom,
! [B3: v,F: $o > v,A: set_o] :
( ( member_v @ B3 @ ( image_o_v @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_450_imageE,axiom,
! [B3: $o,F: v > $o,A: set_v] :
( ( member_o @ B3 @ ( image_v_o @ F @ A ) )
=> ~ ! [X4: v] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_v @ X4 @ A ) ) ) ).
% imageE
thf(fact_451_imageE,axiom,
! [B3: $o,F: $o > $o,A: set_o] :
( ( member_o @ B3 @ ( image_o_o @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_452_imageE,axiom,
! [B3: v,F: set_v > v,A: set_set_v] :
( ( member_v @ B3 @ ( image_set_v_v @ F @ A ) )
=> ~ ! [X4: set_v] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_set_v @ X4 @ A ) ) ) ).
% imageE
thf(fact_453_imageE,axiom,
! [B3: v,F: set_o > v,A: set_set_o] :
( ( member_v @ B3 @ ( image_set_o_v @ F @ A ) )
=> ~ ! [X4: set_o] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_set_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_454_imageE,axiom,
! [B3: $o,F: set_v > $o,A: set_set_v] :
( ( member_o @ B3 @ ( image_set_v_o @ F @ A ) )
=> ~ ! [X4: set_v] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_set_v @ X4 @ A ) ) ) ).
% imageE
thf(fact_455_imageE,axiom,
! [B3: $o,F: set_o > $o,A: set_set_o] :
( ( member_o @ B3 @ ( image_set_o_o @ F @ A ) )
=> ~ ! [X4: set_o] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_set_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_456_imageE,axiom,
! [B3: set_v,F: v > set_v,A: set_v] :
( ( member_set_v @ B3 @ ( image_v_set_v @ F @ A ) )
=> ~ ! [X4: v] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_v @ X4 @ A ) ) ) ).
% imageE
thf(fact_457_imageE,axiom,
! [B3: set_v,F: $o > set_v,A: set_o] :
( ( member_set_v @ B3 @ ( image_o_set_v @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B3
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_458_Compr__image__eq,axiom,
! [F: v > v,A: set_v,P: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ ( image_v_v @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_v_v @ F
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_459_Compr__image__eq,axiom,
! [F: $o > v,A: set_o,P: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ ( image_o_v @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_v @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_460_Compr__image__eq,axiom,
! [F: v > $o,A: set_v,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_v_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_v_o @ F
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_461_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_462_Compr__image__eq,axiom,
! [F: set_o > v,A: set_set_o,P: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ ( image_set_o_v @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_o_v @ F
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_463_Compr__image__eq,axiom,
! [F: set_o > $o,A: set_set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_o_o @ F
@ ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_464_Compr__image__eq,axiom,
! [F: v > set_o,A: set_v,P: set_o > $o] :
( ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ ( image_v_set_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_v_set_o @ F
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_465_Compr__image__eq,axiom,
! [F: $o > set_o,A: set_o,P: set_o > $o] :
( ( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ ( image_o_set_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_set_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_466_Compr__image__eq,axiom,
! [F: set_v > v,A: set_set_v,P: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ ( image_set_v_v @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_v_v @ F
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_467_Compr__image__eq,axiom,
! [F: set_v > $o,A: set_set_v,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_v_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_v_o @ F
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_468_image__Union,axiom,
! [F: v > v,S: set_set_v] :
( ( image_v_v @ F @ ( comple2307003700295860064_set_v @ S ) )
= ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ ( image_v_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_469_image__Union,axiom,
! [F: v > product_prod_v_v,S: set_set_v] :
( ( image_9222788639401671577od_v_v @ F @ ( comple2307003700295860064_set_v @ S ) )
= ( comple5788137035815166516od_v_v @ ( image_7580462938581912729od_v_v @ ( image_9222788639401671577od_v_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_470_image__Union,axiom,
! [F: v > $o,S: set_set_v] :
( ( image_v_o @ F @ ( comple2307003700295860064_set_v @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_v_set_o @ ( image_v_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_471_image__Union,axiom,
! [F: product_prod_v_v > v,S: set_se8455005133513928103od_v_v] :
( ( image_6152814753742948081_v_v_v @ F @ ( comple5788137035815166516od_v_v @ S ) )
= ( comple2307003700295860064_set_v @ ( image_7792316771340735985_set_v @ ( image_6152814753742948081_v_v_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_472_image__Union,axiom,
! [F: product_prod_v_v > product_prod_v_v,S: set_se8455005133513928103od_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( comple5788137035815166516od_v_v @ S ) )
= ( comple5788137035815166516od_v_v @ ( image_5212826947168092101od_v_v @ ( image_781944334261467077od_v_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_473_image__Union,axiom,
! [F: product_prod_v_v > $o,S: set_se8455005133513928103od_v_v] :
( ( image_4100611349647363702_v_v_o @ F @ ( comple5788137035815166516od_v_v @ S ) )
= ( comple90263536869209701_set_o @ ( image_5624318279235061110_set_o @ ( image_4100611349647363702_v_v_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_474_image__Union,axiom,
! [F: $o > v,S: set_set_o] :
( ( image_o_v @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ ( image_o_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_475_image__Union,axiom,
! [F: $o > product_prod_v_v,S: set_set_o] :
( ( image_2239140752724250196od_v_v @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple5788137035815166516od_v_v @ ( image_8240611736229174100od_v_v @ ( image_2239140752724250196od_v_v @ F ) @ S ) ) ) ).
% image_Union
thf(fact_476_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_477_UN__extend__simps_I8_J,axiom,
! [B: v > set_v,A: set_set_v] :
( ( comple2307003700295860064_set_v
@ ( image_set_v_set_v
@ ^ [Y: set_v] : ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ Y ) )
@ A ) )
= ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_478_UN__extend__simps_I8_J,axiom,
! [B: product_prod_v_v > set_v,A: set_se8455005133513928103od_v_v] :
( ( comple2307003700295860064_set_v
@ ( image_7792316771340735985_set_v
@ ^ [Y: set_Product_prod_v_v] : ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ B @ Y ) )
@ A ) )
= ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_479_UN__extend__simps_I8_J,axiom,
! [B: $o > set_v,A: set_set_o] :
( ( comple2307003700295860064_set_v
@ ( image_set_o_set_v
@ ^ [Y: set_o] : ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ Y ) )
@ A ) )
= ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_480_UN__extend__simps_I8_J,axiom,
! [B: v > set_Product_prod_v_v,A: set_set_v] :
( ( comple5788137035815166516od_v_v
@ ( image_7580462938581912729od_v_v
@ ^ [Y: set_v] : ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ B @ Y ) )
@ A ) )
= ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_481_UN__extend__simps_I8_J,axiom,
! [B: product_prod_v_v > set_Product_prod_v_v,A: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v
@ ( image_5212826947168092101od_v_v
@ ^ [Y: set_Product_prod_v_v] : ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ B @ Y ) )
@ A ) )
= ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_482_UN__extend__simps_I8_J,axiom,
! [B: $o > set_Product_prod_v_v,A: set_set_o] :
( ( comple5788137035815166516od_v_v
@ ( image_8240611736229174100od_v_v
@ ^ [Y: set_o] : ( comple5788137035815166516od_v_v @ ( image_2207252236123234868od_v_v @ B @ Y ) )
@ A ) )
= ( comple5788137035815166516od_v_v @ ( image_2207252236123234868od_v_v @ B @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_483_UN__extend__simps_I8_J,axiom,
! [B: v > set_o,A: set_set_v] :
( ( comple90263536869209701_set_o
@ ( image_set_v_set_o
@ ^ [Y: set_v] : ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ Y ) )
@ A ) )
= ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_484_UN__extend__simps_I8_J,axiom,
! [B: product_prod_v_v > set_o,A: set_se8455005133513928103od_v_v] :
( ( comple90263536869209701_set_o
@ ( image_5624318279235061110_set_o
@ ^ [Y: set_Product_prod_v_v] : ( comple90263536869209701_set_o @ ( image_3142478075549801814_set_o @ B @ Y ) )
@ A ) )
= ( comple90263536869209701_set_o @ ( image_3142478075549801814_set_o @ B @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_485_UN__extend__simps_I8_J,axiom,
! [B: $o > set_o,A: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ Y ) )
@ A ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_486_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_487_Sup__SUP__eq,axiom,
( comple8656033219084339145_v_v_o
= ( ^ [S2: set_se4108742076221196076_v_v_o,X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ ( comple6800410395631251988od_v_v @ ( image_9193020243687415530od_v_v @ collec8263177866097347122od_v_v @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_488_Sup__SUP__eq,axiom,
( comple8269352886300210136et_o_o
= ( ^ [S2: set_set_o_o,X2: set_o] : ( member_set_o @ X2 @ ( comple4436988014476444997_set_o @ ( image_4526920052425585868_set_o @ collect_set_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_489_Sup__SUP__eq,axiom,
( comple4059755231828182685et_v_o
= ( ^ [S2: set_set_v_o,X2: set_v] : ( member_set_v @ X2 @ ( comple5450237519782573632_set_v @ ( image_9108510164818287682_set_v @ collect_set_v @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_490_Sup__SUP__eq,axiom,
( complete_Sup_Sup_v_o
= ( ^ [S2: set_v_o,X2: v] : ( member_v @ X2 @ ( comple2307003700295860064_set_v @ ( image_v_o_set_v @ collect_v @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_491_Sup__SUP__eq,axiom,
( comple4105555359904264105_v_v_o
= ( ^ [S2: set_Pr3689507317551669004_v_v_o,X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ ( image_5458630951929891626od_v_v @ collec140062887454715474od_v_v @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_492_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_493_subset__image__iff,axiom,
! [B: set_v,F: v > v,A: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A )
& ( B
= ( image_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_494_subset__image__iff,axiom,
! [B: set_v,F: $o > v,A: set_o] :
( ( ord_less_eq_set_v @ B @ ( image_o_v @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_495_subset__image__iff,axiom,
! [B: set_o,F: v > $o,A: set_v] :
( ( ord_less_eq_set_o @ B @ ( image_v_o @ F @ A ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A )
& ( B
= ( image_v_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_496_subset__image__iff,axiom,
! [B: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_497_subset__image__iff,axiom,
! [B: set_v,F: set_v > v,A: set_set_v] :
( ( ord_less_eq_set_v @ B @ ( image_set_v_v @ F @ A ) )
= ( ? [AA: set_set_v] :
( ( ord_le5216385588623774835_set_v @ AA @ A )
& ( B
= ( image_set_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_498_subset__image__iff,axiom,
! [B: set_o,F: set_v > $o,A: set_set_v] :
( ( ord_less_eq_set_o @ B @ ( image_set_v_o @ F @ A ) )
= ( ? [AA: set_set_v] :
( ( ord_le5216385588623774835_set_v @ AA @ A )
& ( B
= ( image_set_v_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_499_subset__image__iff,axiom,
! [B: set_set_v,F: v > set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ B @ ( image_v_set_v @ F @ A ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A )
& ( B
= ( image_v_set_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_500_subset__image__iff,axiom,
! [B: set_set_v,F: $o > set_v,A: set_o] :
( ( ord_le5216385588623774835_set_v @ B @ ( image_o_set_v @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_set_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_501_subset__image__iff,axiom,
! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A )
& ( B
= ( image_6152814753742948081_v_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_502_subset__image__iff,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A )
& ( B
= ( image_9222788639401671577od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_503_subset__imageE,axiom,
! [B: set_v,F: v > v,A: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
=> ~ ! [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
=> ( B
!= ( image_v_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_504_subset__imageE,axiom,
! [B: set_v,F: $o > v,A: set_o] :
( ( ord_less_eq_set_v @ B @ ( image_o_v @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_505_subset__imageE,axiom,
! [B: set_o,F: v > $o,A: set_v] :
( ( ord_less_eq_set_o @ B @ ( image_v_o @ F @ A ) )
=> ~ ! [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
=> ( B
!= ( image_v_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_506_subset__imageE,axiom,
! [B: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_507_subset__imageE,axiom,
! [B: set_v,F: set_v > v,A: set_set_v] :
( ( ord_less_eq_set_v @ B @ ( image_set_v_v @ F @ A ) )
=> ~ ! [C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ A )
=> ( B
!= ( image_set_v_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_508_subset__imageE,axiom,
! [B: set_o,F: set_v > $o,A: set_set_v] :
( ( ord_less_eq_set_o @ B @ ( image_set_v_o @ F @ A ) )
=> ~ ! [C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ A )
=> ( B
!= ( image_set_v_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_509_subset__imageE,axiom,
! [B: set_set_v,F: v > set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ B @ ( image_v_set_v @ F @ A ) )
=> ~ ! [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
=> ( B
!= ( image_v_set_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_510_subset__imageE,axiom,
! [B: set_set_v,F: $o > set_v,A: set_o] :
( ( ord_le5216385588623774835_set_v @ B @ ( image_o_set_v @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_set_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_511_subset__imageE,axiom,
! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
=> ~ ! [C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ A )
=> ( B
!= ( image_6152814753742948081_v_v_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_512_subset__imageE,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
=> ~ ! [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
=> ( B
!= ( image_9222788639401671577od_v_v @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_513_image__subsetI,axiom,
! [A: set_v,F: v > v,B: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( member_v @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_514_image__subsetI,axiom,
! [A: set_o,F: $o > v,B: set_v] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_v @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_o_v @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_515_image__subsetI,axiom,
! [A: set_v,F: v > $o,B: set_o] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_v_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_516_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_517_image__subsetI,axiom,
! [A: set_v,F: v > set_o,B: set_set_o] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( member_set_o @ ( F @ X4 ) @ B ) )
=> ( ord_le4374716579403074808_set_o @ ( image_v_set_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_518_image__subsetI,axiom,
! [A: set_o,F: $o > set_o,B: set_set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_set_o @ ( F @ X4 ) @ B ) )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_519_image__subsetI,axiom,
! [A: set_set_v,F: set_v > v,B: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( member_v @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_set_v_v @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_520_image__subsetI,axiom,
! [A: set_set_o,F: set_o > v,B: set_v] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( member_v @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_set_o_v @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_521_image__subsetI,axiom,
! [A: set_set_v,F: set_v > $o,B: set_o] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_v_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_522_image__subsetI,axiom,
! [A: set_set_o,F: set_o > $o,B: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_523_image__mono,axiom,
! [A: set_v,B: set_v,F: v > v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A ) @ ( image_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_524_image__mono,axiom,
! [A: set_v,B: set_v,F: v > $o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_less_eq_set_o @ ( image_v_o @ F @ A ) @ ( image_v_o @ F @ B ) ) ) ).
% image_mono
thf(fact_525_image__mono,axiom,
! [A: set_o,B: set_o,F: $o > v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_set_v @ ( image_o_v @ F @ A ) @ ( image_o_v @ F @ B ) ) ) ).
% image_mono
thf(fact_526_image__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ ( image_o_o @ F @ B ) ) ) ).
% image_mono
thf(fact_527_image__mono,axiom,
! [A: set_v,B: set_v,F: v > set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_le5216385588623774835_set_v @ ( image_v_set_v @ F @ A ) @ ( image_v_set_v @ F @ B ) ) ) ).
% image_mono
thf(fact_528_image__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_le5216385588623774835_set_v @ ( image_o_set_v @ F @ A ) @ ( image_o_set_v @ F @ B ) ) ) ).
% image_mono
thf(fact_529_image__mono,axiom,
! [A: set_set_v,B: set_set_v,F: set_v > v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_v @ ( image_set_v_v @ F @ A ) @ ( image_set_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_530_image__mono,axiom,
! [A: set_set_v,B: set_set_v,F: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_less_eq_set_o @ ( image_set_v_o @ F @ A ) @ ( image_set_v_o @ F @ B ) ) ) ).
% image_mono
thf(fact_531_image__mono,axiom,
! [A: set_v,B: set_v,F: v > product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_532_image__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_533_SUP__cong,axiom,
! [A: set_v,B: set_v,C: v > $o,D: v > $o] :
( ( A = B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_v_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_534_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > $o,D: $o > $o] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_535_SUP__cong,axiom,
! [A: set_v,B: set_v,C: v > set_v,D: v > set_v] :
( ( A = B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ C @ A ) )
= ( comple2307003700295860064_set_v @ ( image_v_set_v @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_536_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > set_v,D: $o > set_v] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_o_set_v @ C @ A ) )
= ( comple2307003700295860064_set_v @ ( image_o_set_v @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_537_SUP__cong,axiom,
! [A: set_set_v,B: set_set_v,C: set_v > $o,D: set_v > $o] :
( ( A = B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_v_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_v_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_538_SUP__cong,axiom,
! [A: set_set_o,B: set_set_o,C: set_o > $o,D: set_o > $o] :
( ( A = B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_o_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_o_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_539_SUP__cong,axiom,
! [A: set_v,B: set_v,C: v > set_o,D: v > set_o] :
( ( A = B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_v_set_o @ C @ A ) )
= ( comple90263536869209701_set_o @ ( image_v_set_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_540_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > set_o,D: $o > set_o] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ A ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_541_SUP__cong,axiom,
! [A: set_set_v,B: set_set_v,C: set_v > set_v,D: set_v > set_v] :
( ( A = B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ C @ A ) )
= ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_542_SUP__cong,axiom,
! [A: set_set_o,B: set_set_o,C: set_o > set_v,D: set_o > set_v] :
( ( A = B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ C @ A ) )
= ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_543_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > set_o,B: set_set_o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member_set_o @ ( F @ X4 ) @ B ) )
=> ( ord_le4374716579403074808_set_o @ ( image_set_v_set_o @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_544_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > v,B: set_v] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member_v @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_set_v_v @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_545_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member7453568604450474000od_v_v @ ( F @ X4 ) @ B ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_4094015956495322297od_v_v @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_546_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > $o,B: set_o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_v_o @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_547_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > set_v,B: set_set_v] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member_set_v @ ( F @ X4 ) @ B ) )
=> ( ord_le5216385588623774835_set_v @ ( image_set_v_set_v @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_548_image__Collect__subsetI,axiom,
! [P: set_v > $o,F: set_v > set_Product_prod_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( member8406446414694345712od_v_v @ ( F @ X4 ) @ B ) )
=> ( ord_le4714265922333009223od_v_v @ ( image_7580462938581912729od_v_v @ F @ ( collect_set_v @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_549_setcompr__eq__image,axiom,
! [F: set_v > set_v,P: set_v > $o] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_v] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_set_v_set_v @ F @ ( collect_set_v @ P ) ) ) ).
% setcompr_eq_image
thf(fact_550_Setcompr__eq__image,axiom,
! [F: v > set_v,A: set_v] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v] :
( ( Uu
= ( F @ X2 ) )
& ( member_v @ X2 @ A ) ) )
= ( image_v_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_551_Setcompr__eq__image,axiom,
! [F: $o > set_v,A: set_o] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: $o] :
( ( Uu
= ( F @ X2 ) )
& ( member_o @ X2 @ A ) ) )
= ( image_o_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_552_Setcompr__eq__image,axiom,
! [F: product_prod_v_v > set_v,A: set_Product_prod_v_v] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: product_prod_v_v] :
( ( Uu
= ( F @ X2 ) )
& ( member7453568604450474000od_v_v @ X2 @ A ) ) )
= ( image_2529437795422174673_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_553_Setcompr__eq__image,axiom,
! [F: set_v > set_v,A: set_set_v] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_v] :
( ( Uu
= ( F @ X2 ) )
& ( member_set_v @ X2 @ A ) ) )
= ( image_set_v_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_554_Setcompr__eq__image,axiom,
! [F: set_Product_prod_v_v > set_v,A: set_se8455005133513928103od_v_v] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_Product_prod_v_v] :
( ( Uu
= ( F @ X2 ) )
& ( member8406446414694345712od_v_v @ X2 @ A ) ) )
= ( image_7792316771340735985_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_555_Setcompr__eq__image,axiom,
! [F: set_o > set_v,A: set_set_o] :
( ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_o] :
( ( Uu
= ( F @ X2 ) )
& ( member_set_o @ X2 @ A ) ) )
= ( image_set_o_set_v @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_556_Sup__set__def,axiom,
( comple6800410395631251988od_v_v
= ( ^ [A3: set_se2157405561750842759od_v_v] :
( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] : ( complete_Sup_Sup_o @ ( image_2328747007768699574_v_v_o @ ( member8406446414694345712od_v_v @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_557_Sup__set__def,axiom,
( comple4436988014476444997_set_o
= ( ^ [A3: set_set_set_o] :
( collect_set_o
@ ^ [X2: set_o] : ( complete_Sup_Sup_o @ ( image_set_set_o_o @ ( member_set_o @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_558_Sup__set__def,axiom,
( comple5450237519782573632_set_v
= ( ^ [A3: set_set_set_v] :
( collect_set_v
@ ^ [X2: set_v] : ( complete_Sup_Sup_o @ ( image_set_set_v_o @ ( member_set_v @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_559_Sup__set__def,axiom,
( comple2307003700295860064_set_v
= ( ^ [A3: set_set_v] :
( collect_v
@ ^ [X2: v] : ( complete_Sup_Sup_o @ ( image_set_v_o @ ( member_v @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_560_Sup__set__def,axiom,
( comple5788137035815166516od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( complete_Sup_Sup_o @ ( image_5872996755729839766_v_v_o @ ( member7453568604450474000od_v_v @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_561_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_562_SUP__Sup__eq,axiom,
! [S: set_set_set_v] :
( ( comple4059755231828182685et_v_o
@ ( image_3512364861651339864et_v_o
@ ^ [I: set_set_v,X2: set_v] : ( member_set_v @ X2 @ I )
@ S ) )
= ( ^ [X2: set_v] : ( member_set_v @ X2 @ ( comple5450237519782573632_set_v @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_563_SUP__Sup__eq,axiom,
! [S: set_se2157405561750842759od_v_v] :
( ( comple8656033219084339145_v_v_o
@ ( image_456204219751320920_v_v_o
@ ^ [I: set_se8455005133513928103od_v_v,X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ I )
@ S ) )
= ( ^ [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ ( comple6800410395631251988od_v_v @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_564_SUP__Sup__eq,axiom,
! [S: set_set_set_o] :
( ( comple8269352886300210136et_o_o
@ ( image_5739835251314486680et_o_o
@ ^ [I: set_set_o,X2: set_o] : ( member_set_o @ X2 @ I )
@ S ) )
= ( ^ [X2: set_o] : ( member_set_o @ X2 @ ( comple4436988014476444997_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_565_SUP__Sup__eq,axiom,
! [S: set_set_v] :
( ( complete_Sup_Sup_v_o
@ ( image_set_v_v_o
@ ^ [I: set_v,X2: v] : ( member_v @ X2 @ I )
@ S ) )
= ( ^ [X2: v] : ( member_v @ X2 @ ( comple2307003700295860064_set_v @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_566_SUP__Sup__eq,axiom,
! [S: set_se8455005133513928103od_v_v] :
( ( comple4105555359904264105_v_v_o
@ ( image_5079793815939040600_v_v_o
@ ^ [I: set_Product_prod_v_v,X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ I )
@ S ) )
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ ( comple5788137035815166516od_v_v @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_567_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X2: $o] : ( member_o @ X2 @ I )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_568_UN__E,axiom,
! [B3: v,B: v > set_v,A: set_v] :
( ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) )
=> ~ ! [X4: v] :
( ( member_v @ X4 @ A )
=> ~ ( member_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_569_UN__E,axiom,
! [B3: v,B: $o > set_v,A: set_o] :
( ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_570_UN__E,axiom,
! [B3: $o,B: v > set_o,A: set_v] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) )
=> ~ ! [X4: v] :
( ( member_v @ X4 @ A )
=> ~ ( member_o @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_571_UN__E,axiom,
! [B3: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_572_UN__E,axiom,
! [B3: set_v,B: v > set_set_v,A: set_v] :
( ( member_set_v @ B3 @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ B @ A ) ) )
=> ~ ! [X4: v] :
( ( member_v @ X4 @ A )
=> ~ ( member_set_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_573_UN__E,axiom,
! [B3: set_v,B: $o > set_set_v,A: set_o] :
( ( member_set_v @ B3 @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_set_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_574_UN__E,axiom,
! [B3: set_o,B: v > set_set_o,A: set_v] :
( ( member_set_o @ B3 @ ( comple4436988014476444997_set_o @ ( image_v_set_set_o @ B @ A ) ) )
=> ~ ! [X4: v] :
( ( member_v @ X4 @ A )
=> ~ ( member_set_o @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_575_UN__E,axiom,
! [B3: set_o,B: $o > set_set_o,A: set_o] :
( ( member_set_o @ B3 @ ( comple4436988014476444997_set_o @ ( image_o_set_set_o @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_set_o @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_576_UN__E,axiom,
! [B3: v,B: set_v > set_v,A: set_set_v] :
( ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ B @ A ) ) )
=> ~ ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ~ ( member_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_577_UN__E,axiom,
! [B3: v,B: set_o > set_v,A: set_set_o] :
( ( member_v @ B3 @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ B @ A ) ) )
=> ~ ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ~ ( member_v @ B3 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_578_SUP__eq,axiom,
! [A: set_v,B: set_v,F: v > $o,G: v > $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: v] :
( ( member_v @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: v] :
( ( member_v @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_v_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_579_SUP__eq,axiom,
! [A: set_v,B: set_o,F: v > $o,G: $o > $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_580_SUP__eq,axiom,
! [A: set_o,B: set_v,F: $o > $o,G: v > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X: v] :
( ( member_v @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: v] :
( ( member_v @ J @ B )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_v_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_581_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_582_SUP__eq,axiom,
! [A: set_v,B: set_v,F: v > set_v,G: v > set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: v] :
( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: v] :
( ( member_v @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_set_v @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
= ( comple2307003700295860064_set_v @ ( image_v_set_v @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_583_SUP__eq,axiom,
! [A: set_v,B: set_o,F: v > set_v,G: $o > set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B )
& ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_set_v @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
= ( comple2307003700295860064_set_v @ ( image_o_set_v @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_584_SUP__eq,axiom,
! [A: set_o,B: set_v,F: $o > set_v,G: v > set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X: v] :
( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: v] :
( ( member_v @ J @ B )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_set_v @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) )
= ( comple2307003700295860064_set_v @ ( image_v_set_v @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_585_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > set_v,G: $o > set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B )
& ( ord_less_eq_set_v @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_set_v @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) )
= ( comple2307003700295860064_set_v @ ( image_o_set_v @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_586_SUP__eq,axiom,
! [A: set_v,B: set_set_v,F: v > $o,G: set_v > $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: set_v] :
( ( member_set_v @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: set_v] :
( ( member_set_v @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_v_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_587_SUP__eq,axiom,
! [A: set_v,B: set_set_o,F: v > $o,G: set_o > $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ? [X: set_o] :
( ( member_set_o @ X @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: set_o] :
( ( member_set_o @ J @ B )
=> ? [X: v] :
( ( member_v @ X @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_588_SUP__upper2,axiom,
! [I3: v,A: set_v,U: $o,F: v > $o] :
( ( member_v @ I3 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_589_SUP__upper2,axiom,
! [I3: $o,A: set_o,U: $o,F: $o > $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_590_SUP__upper2,axiom,
! [I3: v,A: set_v,U: set_v,F: v > set_v] :
( ( member_v @ I3 @ A )
=> ( ( ord_less_eq_set_v @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_591_SUP__upper2,axiom,
! [I3: $o,A: set_o,U: set_v,F: $o > set_v] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_set_v @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_592_SUP__upper2,axiom,
! [I3: set_v,A: set_set_v,U: $o,F: set_v > $o] :
( ( member_set_v @ I3 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_v_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_593_SUP__upper2,axiom,
! [I3: set_o,A: set_set_o,U: $o,F: set_o > $o] :
( ( member_set_o @ I3 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_594_SUP__upper2,axiom,
! [I3: v,A: set_v,U: set_o,F: v > set_o] :
( ( member_v @ I3 @ A )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_595_SUP__upper2,axiom,
! [I3: $o,A: set_o,U: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_596_SUP__upper2,axiom,
! [I3: v,A: set_v,U: set_set_v,F: v > set_set_v] :
( ( member_v @ I3 @ A )
=> ( ( ord_le5216385588623774835_set_v @ U @ ( F @ I3 ) )
=> ( ord_le5216385588623774835_set_v @ U @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_597_SUP__upper2,axiom,
! [I3: $o,A: set_o,U: set_set_v,F: $o > set_set_v] :
( ( member_o @ I3 @ A )
=> ( ( ord_le5216385588623774835_set_v @ U @ ( F @ I3 ) )
=> ( ord_le5216385588623774835_set_v @ U @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_598_SUP__upper,axiom,
! [I3: v,A: set_v,F: v > $o] :
( ( member_v @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_599_SUP__upper,axiom,
! [I3: $o,A: set_o,F: $o > $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_600_SUP__upper,axiom,
! [I3: v,A: set_v,F: v > set_v] :
( ( member_v @ I3 @ A )
=> ( ord_less_eq_set_v @ ( F @ I3 ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_601_SUP__upper,axiom,
! [I3: $o,A: set_o,F: $o > set_v] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_v @ ( F @ I3 ) @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_602_SUP__upper,axiom,
! [I3: set_v,A: set_set_v,F: set_v > $o] :
( ( member_set_v @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_set_v_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_603_SUP__upper,axiom,
! [I3: set_o,A: set_set_o,F: set_o > $o] :
( ( member_set_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_set_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_604_SUP__upper,axiom,
! [I3: v,A: set_v,F: v > set_o] :
( ( member_v @ I3 @ A )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_605_SUP__upper,axiom,
! [I3: $o,A: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_606_SUP__upper,axiom,
! [I3: v,A: set_v,F: v > set_set_v] :
( ( member_v @ I3 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I3 ) @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_607_SUP__upper,axiom,
! [I3: $o,A: set_o,F: $o > set_set_v] :
( ( member_o @ I3 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I3 ) @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_608_SUP__least,axiom,
! [A: set_v,F: v > $o,U: $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_609_SUP__least,axiom,
! [A: set_o,F: $o > $o,U: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_610_SUP__least,axiom,
! [A: set_v,F: v > set_v,U: set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_set_v @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_611_SUP__least,axiom,
! [A: set_o,F: $o > set_v,U: set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_v @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_612_SUP__least,axiom,
! [A: set_set_v,F: set_v > $o,U: $o] :
( ! [I2: set_v] :
( ( member_set_v @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_v_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_613_SUP__least,axiom,
! [A: set_set_o,F: set_o > $o,U: $o] :
( ! [I2: set_o] :
( ( member_set_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_614_SUP__least,axiom,
! [A: set_v,F: v > set_o,U: set_o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_615_SUP__least,axiom,
! [A: set_o,F: $o > set_o,U: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_616_SUP__least,axiom,
! [A: set_v,F: v > set_set_v,U: set_set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I2 ) @ U ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_617_SUP__least,axiom,
! [A: set_o,F: $o > set_set_v,U: set_set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I2 ) @ U ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_618_SUP__eqI,axiom,
! [A: set_v,F: v > $o,X5: $o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I4: v] :
( ( member_v @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_v_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_619_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X5: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_620_SUP__eqI,axiom,
! [A: set_v,F: v > set_v,X5: set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_set_v @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_v] :
( ! [I4: v] :
( ( member_v @ I4 @ A )
=> ( ord_less_eq_set_v @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_v @ X5 @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_621_SUP__eqI,axiom,
! [A: set_o,F: $o > set_v,X5: set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_v @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_v] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_v @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_v @ X5 @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_622_SUP__eqI,axiom,
! [A: set_set_v,F: set_v > $o,X5: $o] :
( ! [I2: set_v] :
( ( member_set_v @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I4: set_v] :
( ( member_set_v @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_v_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_623_SUP__eqI,axiom,
! [A: set_set_o,F: set_o > $o,X5: $o] :
( ! [I2: set_o] :
( ( member_set_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I4: set_o] :
( ( member_set_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_o_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_624_SUP__eqI,axiom,
! [A: set_v,F: v > set_o,X5: set_o] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_o] :
( ! [I4: v] :
( ( member_v @ I4 @ A )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X5 @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_625_SUP__eqI,axiom,
! [A: set_o,F: $o > set_o,X5: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X5 @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_626_SUP__eqI,axiom,
! [A: set_v,F: v > set_set_v,X5: set_set_v] :
( ! [I2: v] :
( ( member_v @ I2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_set_v] :
( ! [I4: v] :
( ( member_v @ I4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le5216385588623774835_set_v @ X5 @ Y3 ) )
=> ( ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_627_SUP__eqI,axiom,
! [A: set_o,F: $o > set_set_v,X5: set_set_v] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_set_v] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ I4 ) @ Y3 ) )
=> ( ord_le5216385588623774835_set_v @ X5 @ Y3 ) )
=> ( ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_628_UN__upper,axiom,
! [A2: v,A: set_v,B: v > set_v] :
( ( member_v @ A2 @ A )
=> ( ord_less_eq_set_v @ ( B @ A2 ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_629_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_v] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_v @ ( B @ A2 ) @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_630_UN__upper,axiom,
! [A2: v,A: set_v,B: v > set_o] :
( ( member_v @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_631_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_632_UN__upper,axiom,
! [A2: v,A: set_v,B: v > set_set_v] :
( ( member_v @ A2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( B @ A2 ) @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_633_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_set_v] :
( ( member_o @ A2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( B @ A2 ) @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_634_UN__upper,axiom,
! [A2: set_v,A: set_set_v,B: set_v > set_v] :
( ( member_set_v @ A2 @ A )
=> ( ord_less_eq_set_v @ ( B @ A2 ) @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_635_UN__upper,axiom,
! [A2: set_o,A: set_set_o,B: set_o > set_v] :
( ( member_set_o @ A2 @ A )
=> ( ord_less_eq_set_v @ ( B @ A2 ) @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_636_UN__upper,axiom,
! [A2: set_v,A: set_set_v,B: set_v > set_o] :
( ( member_set_v @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_set_v_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_637_UN__upper,axiom,
! [A2: set_o,A: set_set_o,B: set_o > set_o] :
( ( member_set_o @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_set_o_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_638_UN__least,axiom,
! [A: set_v,B: v > set_v,C: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_639_UN__least,axiom,
! [A: set_o,B: $o > set_v,C: set_v] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_640_UN__least,axiom,
! [A: set_v,B: v > set_o,C: set_o] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_o @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_641_UN__least,axiom,
! [A: set_o,B: $o > set_o,C: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_642_UN__least,axiom,
! [A: set_v,B: v > set_set_v,C: set_set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_643_UN__least,axiom,
! [A: set_o,B: $o > set_set_v,C: set_set_v] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_644_UN__least,axiom,
! [A: set_set_v,B: set_v > set_v,C: set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_645_UN__least,axiom,
! [A: set_set_o,B: set_o > set_v,C: set_v] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_set_v @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_646_UN__least,axiom,
! [A: set_set_v,B: set_v > set_o,C: set_o] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_o @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_v_set_o @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_647_UN__least,axiom,
! [A: set_set_o,B: set_o > set_o,C: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( B @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_o_set_o @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_648_UN__mono,axiom,
! [A: set_v,B: set_v,F: v > set_v,G: v > set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_649_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_v,G: $o > set_v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_650_UN__mono,axiom,
! [A: set_v,B: set_v,F: v > set_o,G: v > set_o] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_v_set_o @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_651_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_652_UN__mono,axiom,
! [A: set_v,B: set_v,F: v > set_set_v,G: v > set_set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ F @ A ) ) @ ( comple5450237519782573632_set_v @ ( image_v_set_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_653_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_set_v,G: $o > set_set_v] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ F @ A ) ) @ ( comple5450237519782573632_set_v @ ( image_o_set_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_654_UN__mono,axiom,
! [A: set_set_o,B: set_set_o,F: set_o > set_v,G: set_o > set_v] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_655_UN__mono,axiom,
! [A: set_set_v,B: set_set_v,F: set_v > set_v,G: set_v > set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ F @ A ) ) @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_656_UN__mono,axiom,
! [A: set_set_o,B: set_set_o,F: set_o > set_o,G: set_o > set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_o_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_set_o_set_o @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_657_UN__mono,axiom,
! [A: set_set_v,B: set_set_v,F: set_v > set_o,G: set_v > set_o] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_set_v_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_set_v_set_o @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_658_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_659_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_660_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_661_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y3 )
=> ~ ( member_v @ A22 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_662_re__refl,axiom,
! [X5: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ X5 ) ).
% re_refl
thf(fact_663_re__succ,axiom,
! [X5: v,Y2: v,Z2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y2 )
=> ( ( member_v @ Z2 @ ( successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Z2 ) ) ) ).
% re_succ
thf(fact_664_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y )
& ( member_v @ Z5 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.simps
thf(fact_665_succ__re,axiom,
! [Y2: v,X5: v,Z2: v] :
( ( member_v @ Y2 @ ( successors @ X5 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Z2 ) ) ) ).
% succ_re
thf(fact_666_all__subset__image,axiom,
! [F: v > v,A: set_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( P @ ( image_v_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_667_all__subset__image,axiom,
! [F: $o > v,A: set_o,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_o_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( P @ ( image_o_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_668_all__subset__image,axiom,
! [F: v > $o,A: set_v,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_v_o @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( P @ ( image_v_o @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_669_all__subset__image,axiom,
! [F: $o > $o,A: set_o,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( P @ ( image_o_o @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_670_all__subset__image,axiom,
! [F: set_v > v,A: set_set_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_set_v_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( P @ ( image_set_v_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_671_all__subset__image,axiom,
! [F: set_v > $o,A: set_set_v,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_set_v_o @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( P @ ( image_set_v_o @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_672_all__subset__image,axiom,
! [F: v > set_v,A: set_v,P: set_set_v > $o] :
( ( ! [B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ ( image_v_set_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( P @ ( image_v_set_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_673_all__subset__image,axiom,
! [F: $o > set_v,A: set_o,P: set_set_v > $o] :
( ( ! [B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ ( image_o_set_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( P @ ( image_o_set_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_674_all__subset__image,axiom,
! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_675_all__subset__image,axiom,
! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_676_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_677_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_678_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_679_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_680_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_681_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_682_graph_Ounite__sub__env,axiom,
! [Vertices: set_o,Successors: $o > set_o,V: $o,E: sCC_Bl3933283745082679452t_unit,W: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl701244671244948650t_unit @ Successors @ V @ E )
=> ( ( member_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl2127731265848312895t_unit @ E @ V ) )
=> ( ( member_o @ W @ ( sCC_Bl3035372653189477680t_unit @ E ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl4316552501791471509t_unit @ E ) )
=> ( sCC_Bl1874319547970246138t_unit @ E @ ( sCC_Bloemen_unite_o @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_683_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_684_graph_Ounite__sub__env,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,V: set_v,E: sCC_Bl337355980704484737t_unit,W: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7011918439528173327t_unit @ Successors @ V @ E )
=> ( ( member_set_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl1947314421851746340t_unit @ E @ V ) )
=> ( ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl8056216847179350138t_unit @ E ) )
=> ( sCC_Bl4683737126326172629t_unit @ E @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_685_graph_Ounite__sub__env,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,V: set_Product_prod_v_v,E: sCC_Bl471817294227068333t_unit,W: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl104600603521692219t_unit @ Successors @ V @ E )
=> ( ( member8406446414694345712od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl454723783576663632t_unit @ E @ V ) )
=> ( ( member8406446414694345712od_v_v @ W @ ( sCC_Bl647639559028580801t_unit @ E ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl2694530823580680358t_unit @ E ) )
=> ( sCC_Bl6409147186158772137t_unit @ E @ ( sCC_Bl2743269859033768757od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_686_graph_Ounite__sub__env,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,V: set_o,E: sCC_Bl8746163187549392316t_unit,W: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl5138946541064818122t_unit @ Successors @ V @ E )
=> ( ( member_set_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl5177287912846518623t_unit @ E @ V ) )
=> ( ( member_set_o @ W @ ( sCC_Bl8197997162987607632t_unit @ E ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl4929516568782881973t_unit @ E ) )
=> ( sCC_Bl2819566960983106778t_unit @ E @ ( sCC_Bl5184775513941676262_set_o @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_687_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_688_UN__constant__eq,axiom,
! [A2: v,A: set_v,F: v > set_v,C2: set_v] :
( ( member_v @ A2 @ A )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_689_UN__constant__eq,axiom,
! [A2: $o,A: set_o,F: $o > set_v,C2: set_v] :
( ( member_o @ A2 @ A )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_o_set_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_690_UN__constant__eq,axiom,
! [A2: v,A: set_v,F: v > set_o,C2: set_o] :
( ( member_v @ A2 @ A )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_691_UN__constant__eq,axiom,
! [A2: $o,A: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A2 @ A )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_692_UN__constant__eq,axiom,
! [A2: set_v,A: set_set_v,F: set_v > set_v,C2: set_v] :
( ( member_set_v @ A2 @ A )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_693_UN__constant__eq,axiom,
! [A2: set_o,A: set_set_o,F: set_o > set_v,C2: set_v] :
( ( member_set_o @ A2 @ A )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_694_UN__constant__eq,axiom,
! [A2: set_v,A: set_set_v,F: set_v > set_o,C2: set_o] :
( ( member_set_v @ A2 @ A )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_set_v_set_o @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_695_UN__constant__eq,axiom,
! [A2: set_o,A: set_set_o,F: set_o > set_o,C2: set_o] :
( ( member_set_o @ A2 @ A )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_set_o_set_o @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_696_UN__constant__eq,axiom,
! [A2: product_prod_v_v,A: set_Product_prod_v_v,F: product_prod_v_v > set_v,C2: set_v] :
( ( member7453568604450474000od_v_v @ A2 @ A )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_697_UN__constant__eq,axiom,
! [A2: v,A: set_v,F: v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( member_v @ A2 @ A )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ A ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_698_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_699_reachable__re,axiom,
! [X5: v,Y2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y2 ) ) ).
% reachable_re
thf(fact_700_re__reachable,axiom,
! [X5: v,Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X5 @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 ) ) ).
% re_reachable
thf(fact_701_ra__trans,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z2 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Z2 @ E4 ) ) ) ).
% ra_trans
thf(fact_702_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_703_reachable__refl,axiom,
! [X5: v] : ( sCC_Bl649662514949026229able_v @ successors @ X5 @ X5 ) ).
% reachable_refl
thf(fact_704_reachable__succ,axiom,
! [Y2: v,X5: v,Z2: v] :
( ( member_v @ Y2 @ ( successors @ X5 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% reachable_succ
thf(fact_705_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_v @ Y @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z5 ) ) ) ) ).
% reachable.simps
thf(fact_706_reachable__edge,axiom,
! [Y2: v,X5: v] :
( ( member_v @ Y2 @ ( successors @ X5 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 ) ) ).
% reachable_edge
thf(fact_707_reachable__end__induct,axiom,
! [X5: v,Y2: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y3: v,Z6: v] :
( ( P @ X4 @ Y3 )
=> ( ( member_v @ Z6 @ ( successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ).
% reachable_end_induct
thf(fact_708_reachable__trans,axiom,
! [X5: v,Y2: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% reachable_trans
thf(fact_709_succ__reachable,axiom,
! [X5: v,Y2: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( ( member_v @ Z2 @ ( successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Z2 ) ) ) ).
% succ_reachable
thf(fact_710_ra__mono,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E5 ) ) ) ).
% ra_mono
thf(fact_711_ra__refl,axiom,
! [X5: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ X5 @ E4 ) ).
% ra_refl
thf(fact_712_ra__reachable,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 ) ) ).
% ra_reachable
thf(fact_713_finite__Collect__conjI,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ( finite_finite_set_v @ ( collect_set_v @ P ) )
| ( finite_finite_set_v @ ( collect_set_v @ Q ) ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_714_finite__Collect__conjI,axiom,
! [P: v > $o,Q: v > $o] :
( ( ( finite_finite_v @ ( collect_v @ P ) )
| ( finite_finite_v @ ( collect_v @ Q ) ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_715_finite__Collect__disjI,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_set_v @ ( collect_set_v @ P ) )
& ( finite_finite_set_v @ ( collect_set_v @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_716_finite__Collect__disjI,axiom,
! [P: v > $o,Q: v > $o] :
( ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_v @ ( collect_v @ P ) )
& ( finite_finite_v @ ( collect_v @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_717_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_718_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_719_finite__imageI,axiom,
! [F3: set_v,H: v > v] :
( ( finite_finite_v @ F3 )
=> ( finite_finite_v @ ( image_v_v @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_720_finite__Union,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ! [M: set_v] :
( ( member_set_v @ M @ A )
=> ( finite_finite_v @ M ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% finite_Union
thf(fact_721_finite__Union,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ! [M: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ M @ A )
=> ( finite3348123685078250256od_v_v @ M ) )
=> ( finite3348123685078250256od_v_v @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% finite_Union
thf(fact_722_finite__Union,axiom,
! [A: set_set_o] :
( ( finite_finite_set_o @ A )
=> ( ! [M: set_o] :
( ( member_set_o @ M @ A )
=> ( finite_finite_o @ M ) )
=> ( finite_finite_o @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% finite_Union
thf(fact_723_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_724_finite__Collect__subsets,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [B2: set_v] : ( ord_less_eq_set_v @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_725_finite__Collect__subsets,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( finite6084192165098772208od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_726_finite__Collect__subsets,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( finite_finite_set_o
@ ( collect_set_o
@ ^ [B2: set_o] : ( ord_less_eq_set_o @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_727_finite__Collect__subsets,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( finite8701002811114149628_set_v
@ ( collect_set_set_v
@ ^ [B2: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_728_finite__Collect__subsets,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( finite2944697185239447248od_v_v
@ ( collec5787501045982241298od_v_v
@ ^ [B2: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_729_finite__Collect__bounded__ex,axiom,
! [P: set_v > $o,Q: set_v > set_v > $o] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
? [Y: set_v] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: set_v] :
( ( P @ Y )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_730_finite__Collect__bounded__ex,axiom,
! [P: set_v > $o,Q: v > set_v > $o] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] :
? [Y: set_v] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: set_v] :
( ( P @ Y )
=> ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_731_finite__Collect__bounded__ex,axiom,
! [P: v > $o,Q: set_v > v > $o] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
? [Y: v] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: v] :
( ( P @ Y )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [X2: set_v] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_732_finite__Collect__bounded__ex,axiom,
! [P: v > $o,Q: v > v > $o] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] :
? [Y: v] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: v] :
( ( P @ Y )
=> ( finite_finite_v
@ ( collect_v
@ ^ [X2: v] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_733_pfx_I2_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% pfx(2)
thf(fact_734_finite__UN,axiom,
! [A: set_v,B: v > set_v] :
( ( finite_finite_v @ A )
=> ( ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ( finite_finite_v @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_735_finite__UN,axiom,
! [A: set_v,B: v > set_Product_prod_v_v] :
( ( finite_finite_v @ A )
=> ( ( finite3348123685078250256od_v_v @ ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ B @ A ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ( finite3348123685078250256od_v_v @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_736_finite__UN,axiom,
! [A: set_v,B: v > set_o] :
( ( finite_finite_v @ A )
=> ( ( finite_finite_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A )
=> ( finite_finite_o @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_737_finite__UN__I,axiom,
! [A: set_o,B: $o > set_v] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite_finite_v @ ( B @ A5 ) ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_o_set_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_738_finite__UN__I,axiom,
! [A: set_v,B: v > set_v] :
( ( finite_finite_v @ A )
=> ( ! [A5: v] :
( ( member_v @ A5 @ A )
=> ( finite_finite_v @ ( B @ A5 ) ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_v_set_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_739_finite__UN__I,axiom,
! [A: set_o,B: $o > set_o] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite_finite_o @ ( B @ A5 ) ) )
=> ( finite_finite_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_740_finite__UN__I,axiom,
! [A: set_v,B: v > set_o] :
( ( finite_finite_v @ A )
=> ( ! [A5: v] :
( ( member_v @ A5 @ A )
=> ( finite_finite_o @ ( B @ A5 ) ) )
=> ( finite_finite_o @ ( comple90263536869209701_set_o @ ( image_v_set_o @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_741_finite__UN__I,axiom,
! [A: set_set_v,B: set_v > set_v] :
( ( finite_finite_set_v @ A )
=> ( ! [A5: set_v] :
( ( member_set_v @ A5 @ A )
=> ( finite_finite_v @ ( B @ A5 ) ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_set_v_set_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_742_finite__UN__I,axiom,
! [A: set_set_o,B: set_o > set_v] :
( ( finite_finite_set_o @ A )
=> ( ! [A5: set_o] :
( ( member_set_o @ A5 @ A )
=> ( finite_finite_v @ ( B @ A5 ) ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_set_o_set_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_743_finite__UN__I,axiom,
! [A: set_set_v,B: set_v > set_o] :
( ( finite_finite_set_v @ A )
=> ( ! [A5: set_v] :
( ( member_set_v @ A5 @ A )
=> ( finite_finite_o @ ( B @ A5 ) ) )
=> ( finite_finite_o @ ( comple90263536869209701_set_o @ ( image_set_v_set_o @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_744_finite__UN__I,axiom,
! [A: set_set_o,B: set_o > set_o] :
( ( finite_finite_set_o @ A )
=> ( ! [A5: set_o] :
( ( member_set_o @ A5 @ A )
=> ( finite_finite_o @ ( B @ A5 ) ) )
=> ( finite_finite_o @ ( comple90263536869209701_set_o @ ( image_set_o_set_o @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_745_finite__UN__I,axiom,
! [A: set_Product_prod_v_v,B: product_prod_v_v > set_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ! [A5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A5 @ A )
=> ( finite_finite_v @ ( B @ A5 ) ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_746_finite__UN__I,axiom,
! [A: set_o,B: $o > set_Product_prod_v_v] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite3348123685078250256od_v_v @ ( B @ A5 ) ) )
=> ( finite3348123685078250256od_v_v @ ( comple5788137035815166516od_v_v @ ( image_2207252236123234868od_v_v @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_747_sccE,axiom,
! [S: set_v,X5: v,Y2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X5 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X5 )
=> ( member_v @ Y2 @ S ) ) ) ) ) ).
% sccE
thf(fact_748_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E6: sCC_Bl1394983891496994913t_unit,E7: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E7 )
= ( sCC_Bl1090238580953940555t_unit @ E6 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E6 ) @ ( sCC_Bl4645233313691564917t_unit @ E7 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E6 ) @ ( sCC_Bl157864678168468314t_unit @ E7 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E6 @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E7 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E6 @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E7 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E6 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E6 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E7 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E7 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_749_finite__UnionD,axiom,
! [A: set_set_v] :
( ( finite_finite_v @ ( comple2307003700295860064_set_v @ A ) )
=> ( finite_finite_set_v @ A ) ) ).
% finite_UnionD
thf(fact_750_finite__UnionD,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite3348123685078250256od_v_v @ ( comple5788137035815166516od_v_v @ A ) )
=> ( finite6084192165098772208od_v_v @ A ) ) ).
% finite_UnionD
thf(fact_751_finite__UnionD,axiom,
! [A: set_set_o] :
( ( finite_finite_o @ ( comple90263536869209701_set_o @ A ) )
=> ( finite_finite_set_o @ A ) ) ).
% finite_UnionD
thf(fact_752_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_v,R: $o > v > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_753_pigeonhole__infinite__rel,axiom,
! [A: set_Product_prod_v_v,B: set_v,R: product_prod_v_v > v > $o] :
( ~ ( finite3348123685078250256od_v_v @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite3348123685078250256od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_754_pigeonhole__infinite__rel,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_v,R: set_Product_prod_v_v > v > $o] :
( ~ ( finite6084192165098772208od_v_v @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite6084192165098772208od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [A4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_755_pigeonhole__infinite__rel,axiom,
! [A: set_set_o,B: set_v,R: set_o > v > $o] :
( ~ ( finite_finite_set_o @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite_finite_set_o
@ ( collect_set_o
@ ^ [A4: set_o] :
( ( member_set_o @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_756_pigeonhole__infinite__rel,axiom,
! [A: set_set_v,B: set_v,R: set_v > v > $o] :
( ~ ( finite_finite_set_v @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite_finite_set_v
@ ( collect_set_v
@ ^ [A4: set_v] :
( ( member_set_v @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_757_pigeonhole__infinite__rel,axiom,
! [A: set_v,B: set_v,R: v > v > $o] :
( ~ ( finite_finite_v @ A )
=> ( ( finite_finite_v @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A )
=> ? [Xa: v] :
( ( member_v @ Xa @ B )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: v] :
( ( member_v @ X4 @ B )
& ~ ( finite_finite_v
@ ( collect_v
@ ^ [A4: v] :
( ( member_v @ A4 @ A )
& ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_758_not__finite__existsD,axiom,
! [P: set_v > $o] :
( ~ ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ? [X_1: set_v] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_759_not__finite__existsD,axiom,
! [P: v > $o] :
( ~ ( finite_finite_v @ ( collect_v @ P ) )
=> ? [X_1: v] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_760_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_re
thf(fact_761_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.re_reachable
thf(fact_762_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_763_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z2 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ Z2 @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_764_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.ra_reachable
thf(fact_765_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_766_graph_Oreachable__edge,axiom,
! [Vertices: set_o,Successors: $o > set_o,Y2: $o,X5: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( member_o @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_767_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X5: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_768_graph_Oreachable__edge,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,Y2: set_v,X5: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( member_set_v @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_769_graph_Oreachable__edge,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,Y2: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( member8406446414694345712od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_770_graph_Oreachable__edge,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,Y2: set_o,X5: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( member_set_o @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_771_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X5: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X5 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_772_graph_Osucc__reachable,axiom,
! [Vertices: set_o,Successors: $o > set_o,X5: $o,Y2: $o,Z2: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Y2 )
=> ( ( member_o @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_773_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X5: product_prod_v_v,Y2: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_774_graph_Osucc__reachable,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,X5: set_v,Y2: set_v,Z2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Y2 )
=> ( ( member_set_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_775_graph_Osucc__reachable,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Y2 )
=> ( ( member8406446414694345712od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_776_graph_Osucc__reachable,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,X5: set_o,Y2: set_o,Z2: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Y2 )
=> ( ( member_set_o @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_777_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 )
=> ( ( member_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_778_graph_Oreachable_Ocases,axiom,
! [Vertices: set_o,Successors: $o > set_o,A1: $o,A22: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ A1 @ A22 )
=> ( ( A22 = ~ A1 )
=> ~ ! [Y3: $o] :
( ( member_o @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl1688025099652275002able_o @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_779_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_780_graph_Oreachable_Ocases,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,A1: set_v,A22: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_v] :
( ( member_set_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl7354734129683093653_set_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_781_graph_Oreachable_Ocases,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,A1: set_Product_prod_v_v,A22: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl8444643370470080105od_v_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_782_graph_Oreachable_Ocases,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,A1: set_o,A22: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_o] :
( ( member_set_o @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4440840191236725274_set_o @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_783_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_784_graph_Oreachable_Osimps,axiom,
! [Vertices: set_o,Successors: $o > set_o,A1: $o,A22: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ A1 @ A22 )
= ( ? [X2: $o] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: $o,Y: $o,Z5: $o] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_o @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl1688025099652275002able_o @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_785_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y: product_prod_v_v,Z5: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member7453568604450474000od_v_v @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_786_graph_Oreachable_Osimps,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,A1: set_v,A22: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ A1 @ A22 )
= ( ? [X2: set_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_v,Y: set_v,Z5: set_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_set_v @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_787_graph_Oreachable_Osimps,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,A1: set_Product_prod_v_v,A22: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z5: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member8406446414694345712od_v_v @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl8444643370470080105od_v_v @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_788_graph_Oreachable_Osimps,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,A1: set_o,A22: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ A1 @ A22 )
= ( ? [X2: set_o] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_o,Y: set_o,Z5: set_o] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_set_o @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl4440840191236725274_set_o @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_789_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_v @ Y @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_790_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_trans
thf(fact_791_graph_Oreachable__end__induct,axiom,
! [Vertices: set_o,Successors: $o > set_o,X5: $o,Y2: $o,P: $o > $o > $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Y2 )
=> ( ! [X4: $o] : ( P @ X4 @ X4 )
=> ( ! [X4: $o,Y3: $o,Z6: $o] :
( ( P @ X4 @ Y3 )
=> ( ( member_o @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_792_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X5: product_prod_v_v,Y2: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Y2 )
=> ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: product_prod_v_v,Y3: product_prod_v_v,Z6: product_prod_v_v] :
( ( P @ X4 @ Y3 )
=> ( ( member7453568604450474000od_v_v @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_793_graph_Oreachable__end__induct,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,X5: set_v,Y2: set_v,P: set_v > set_v > $o] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Y2 )
=> ( ! [X4: set_v] : ( P @ X4 @ X4 )
=> ( ! [X4: set_v,Y3: set_v,Z6: set_v] :
( ( P @ X4 @ Y3 )
=> ( ( member_set_v @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_794_graph_Oreachable__end__induct,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v,P: set_Product_prod_v_v > set_Product_prod_v_v > $o] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Y2 )
=> ( ! [X4: set_Product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z6: set_Product_prod_v_v] :
( ( P @ X4 @ Y3 )
=> ( ( member8406446414694345712od_v_v @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_795_graph_Oreachable__end__induct,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,X5: set_o,Y2: set_o,P: set_o > set_o > $o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Y2 )
=> ( ! [X4: set_o] : ( P @ X4 @ X4 )
=> ( ! [X4: set_o,Y3: set_o,Z6: set_o] :
( ( P @ X4 @ Y3 )
=> ( ( member_set_o @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_796_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y3: v,Z6: v] :
( ( P @ X4 @ Y3 )
=> ( ( member_v @ Z6 @ ( Successors @ Y3 ) )
=> ( P @ X4 @ Z6 ) ) )
=> ( P @ X5 @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_797_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_798_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ X5 ) ) ).
% graph.reachable_refl
thf(fact_799_graph_Oreachable__succ,axiom,
! [Vertices: set_o,Successors: $o > set_o,Y2: $o,X5: $o,Z2: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( member_o @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_800_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X5: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_801_graph_Oreachable__succ,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,Y2: set_v,X5: set_v,Z2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( member_set_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_802_graph_Oreachable__succ,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,Y2: set_Product_prod_v_v,X5: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( member8406446414694345712od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_803_graph_Oreachable__succ,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,Y2: set_o,X5: set_o,Z2: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( member_set_o @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_804_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X5: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_805_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ X5 @ E4 ) ) ).
% graph.ra_refl
thf(fact_806_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_807_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_o
= ( ^ [Vertices2: set_o,Successors2: $o > set_o] :
( ( finite_finite_o @ Vertices2 )
& ! [X2: $o] :
( ( member_o @ X2 @ Vertices2 )
=> ( ord_less_eq_set_o @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_808_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl5810666556806954322_set_v
= ( ^ [Vertices2: set_set_v,Successors2: set_v > set_set_v] :
( ( finite_finite_set_v @ Vertices2 )
& ! [X2: set_v] :
( ( member_set_v @ X2 @ Vertices2 )
=> ( ord_le5216385588623774835_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_809_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8431766404916698662od_v_v
= ( ^ [Vertices2: set_se8455005133513928103od_v_v,Successors2: set_Product_prod_v_v > set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ Vertices2 )
& ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ Vertices2 )
=> ( ord_le4714265922333009223od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_810_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_811_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_812_graph_Ointro,axiom,
! [Vertices: set_o,Successors: $o > set_o] :
( ( finite_finite_o @ Vertices )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ Vertices )
=> ( ord_less_eq_set_o @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_o @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_813_graph_Ointro,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v] :
( ( finite_finite_set_v @ Vertices )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ Vertices )
=> ( ord_le5216385588623774835_set_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_814_graph_Ointro,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ Vertices )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ Vertices )
=> ( ord_le4714265922333009223od_v_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_815_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_816_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X5 @ Y2 @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_817_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_818_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_819_Fpow__def,axiom,
( finite_Fpow_v
= ( ^ [A3: set_v] :
( collect_set_v
@ ^ [X7: set_v] :
( ( ord_less_eq_set_v @ X7 @ A3 )
& ( finite_finite_v @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_820_Fpow__def,axiom,
( finite315275465967970893od_v_v
= ( ^ [A3: set_Product_prod_v_v] :
( collec8263177866097347122od_v_v
@ ^ [X7: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X7 @ A3 )
& ( finite3348123685078250256od_v_v @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_821_Fpow__def,axiom,
( finite_Fpow_o
= ( ^ [A3: set_o] :
( collect_set_o
@ ^ [X7: set_o] :
( ( ord_less_eq_set_o @ X7 @ A3 )
& ( finite_finite_o @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_822_Fpow__def,axiom,
( finite_Fpow_set_v
= ( ^ [A3: set_set_v] :
( collect_set_set_v
@ ^ [X7: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X7 @ A3 )
& ( finite_finite_set_v @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_823_Fpow__def,axiom,
( finite1064079062865762861od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v] :
( collec5787501045982241298od_v_v
@ ^ [X7: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ X7 @ A3 )
& ( finite6084192165098772208od_v_v @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_824_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_825_graph_Osclosed,axiom,
! [Vertices: set_o,Successors: $o > set_o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ! [X: $o] :
( ( member_o @ X @ Vertices )
=> ( ord_less_eq_set_o @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_826_graph_Osclosed,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ! [X: set_v] :
( ( member_set_v @ X @ Vertices )
=> ( ord_le5216385588623774835_set_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_827_graph_Osclosed,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ! [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ Vertices )
=> ( ord_le4714265922333009223od_v_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_828_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X: v] :
( ( member_v @ X @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_829_finite__has__minimal2,axiom,
! [A: set_set_v,A2: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A2 @ A )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A )
& ( ord_less_eq_set_v @ X4 @ A2 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A )
=> ( ( ord_less_eq_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_830_finite__has__minimal2,axiom,
! [A: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A2 @ A )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
& ( ord_le7336532860387713383od_v_v @ X4 @ A2 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_831_finite__has__minimal2,axiom,
! [A: set_set_o,A2: set_o] :
( ( finite_finite_set_o @ A )
=> ( ( member_set_o @ A2 @ A )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A )
& ( ord_less_eq_set_o @ X4 @ A2 )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_832_finite__has__minimal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ X4 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_833_finite__has__minimal2,axiom,
! [A: set_set_set_v,A2: set_set_v] :
( ( finite8701002811114149628_set_v @ A )
=> ( ( member_set_set_v @ A2 @ A )
=> ? [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
& ( ord_le5216385588623774835_set_v @ X4 @ A2 )
& ! [Xa: set_set_v] :
( ( member_set_set_v @ Xa @ A )
=> ( ( ord_le5216385588623774835_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_834_finite__has__minimal2,axiom,
! [A: set_se2157405561750842759od_v_v,A2: set_se8455005133513928103od_v_v] :
( ( finite2944697185239447248od_v_v @ A )
=> ( ( member5511408251247217616od_v_v @ A2 @ A )
=> ? [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
& ( ord_le4714265922333009223od_v_v @ X4 @ A2 )
& ! [Xa: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Xa @ A )
=> ( ( ord_le4714265922333009223od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_835_finite__has__maximal2,axiom,
! [A: set_set_v,A2: set_v] :
( ( finite_finite_set_v @ A )
=> ( ( member_set_v @ A2 @ A )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A )
& ( ord_less_eq_set_v @ A2 @ X4 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A )
=> ( ( ord_less_eq_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_836_finite__has__maximal2,axiom,
! [A: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( member8406446414694345712od_v_v @ A2 @ A )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
& ( ord_le7336532860387713383od_v_v @ A2 @ X4 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_837_finite__has__maximal2,axiom,
! [A: set_set_o,A2: set_o] :
( ( finite_finite_set_o @ A )
=> ( ( member_set_o @ A2 @ A )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A )
& ( ord_less_eq_set_o @ A2 @ X4 )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_838_finite__has__maximal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ A2 @ X4 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_839_finite__has__maximal2,axiom,
! [A: set_set_set_v,A2: set_set_v] :
( ( finite8701002811114149628_set_v @ A )
=> ( ( member_set_set_v @ A2 @ A )
=> ? [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
& ( ord_le5216385588623774835_set_v @ A2 @ X4 )
& ! [Xa: set_set_v] :
( ( member_set_set_v @ Xa @ A )
=> ( ( ord_le5216385588623774835_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_840_finite__has__maximal2,axiom,
! [A: set_se2157405561750842759od_v_v,A2: set_se8455005133513928103od_v_v] :
( ( finite2944697185239447248od_v_v @ A )
=> ( ( member5511408251247217616od_v_v @ A2 @ A )
=> ? [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
& ( ord_le4714265922333009223od_v_v @ A2 @ X4 )
& ! [Xa: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Xa @ A )
=> ( ( ord_le4714265922333009223od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_841_rev__finite__subset,axiom,
! [B: set_v,A: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( finite_finite_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_842_rev__finite__subset,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( finite3348123685078250256od_v_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_843_rev__finite__subset,axiom,
! [B: set_o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( finite_finite_o @ A ) ) ) ).
% rev_finite_subset
thf(fact_844_rev__finite__subset,axiom,
! [B: set_set_v,A: set_set_v] :
( ( finite_finite_set_v @ B )
=> ( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( finite_finite_set_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_845_rev__finite__subset,axiom,
! [B: set_se8455005133513928103od_v_v,A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ B )
=> ( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( finite6084192165098772208od_v_v @ A ) ) ) ).
% rev_finite_subset
thf(fact_846_infinite__super,axiom,
! [S: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_847_infinite__super,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_848_infinite__super,axiom,
! [S: set_o,T2: set_o] :
( ( ord_less_eq_set_o @ S @ T2 )
=> ( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ T2 ) ) ) ).
% infinite_super
thf(fact_849_infinite__super,axiom,
! [S: set_set_v,T2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ S @ T2 )
=> ( ~ ( finite_finite_set_v @ S )
=> ~ ( finite_finite_set_v @ T2 ) ) ) ).
% infinite_super
thf(fact_850_infinite__super,axiom,
! [S: set_se8455005133513928103od_v_v,T2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ S @ T2 )
=> ( ~ ( finite6084192165098772208od_v_v @ S )
=> ~ ( finite6084192165098772208od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_851_finite__subset,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( finite_finite_v @ B )
=> ( finite_finite_v @ A ) ) ) ).
% finite_subset
thf(fact_852_finite__subset,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( finite3348123685078250256od_v_v @ B )
=> ( finite3348123685078250256od_v_v @ A ) ) ) ).
% finite_subset
thf(fact_853_finite__subset,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( finite_finite_o @ B )
=> ( finite_finite_o @ A ) ) ) ).
% finite_subset
thf(fact_854_finite__subset,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( finite_finite_set_v @ B )
=> ( finite_finite_set_v @ A ) ) ) ).
% finite_subset
thf(fact_855_finite__subset,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( finite6084192165098772208od_v_v @ A ) ) ) ).
% finite_subset
thf(fact_856_graph_Osucc__re,axiom,
! [Vertices: set_o,Successors: $o > set_o,Y2: $o,X5: $o,Z2: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( member_o @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl6452251072014827417_end_o @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl6452251072014827417_end_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_857_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X5: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_858_graph_Osucc__re,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,Y2: set_v,X5: set_v,Z2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( member_set_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl1820819216308814196_set_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl1820819216308814196_set_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_859_graph_Osucc__re,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,Y2: set_Product_prod_v_v,X5: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( member8406446414694345712od_v_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl3679296176305320008od_v_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl3679296176305320008od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_860_graph_Osucc__re,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,Y2: set_o,X5: set_o,Z2: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( member_set_o @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl4846409930415309433_set_o @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl4846409930415309433_set_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_861_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X5: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X5 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_862_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_o,Successors: $o > set_o,A1: $o,A22: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl6452251072014827417_end_o @ Successors @ A1 @ A22 )
=> ( ( A22 = ~ A1 )
=> ~ ! [Y3: $o] :
( ( sCC_Bl6452251072014827417_end_o @ Successors @ A1 @ Y3 )
=> ~ ( member_o @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_863_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y3 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_864_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,A1: set_v,A22: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1820819216308814196_set_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_v] :
( ( sCC_Bl1820819216308814196_set_v @ Successors @ A1 @ Y3 )
=> ~ ( member_set_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_865_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,A1: set_Product_prod_v_v,A22: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3679296176305320008od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_Product_prod_v_v] :
( ( sCC_Bl3679296176305320008od_v_v @ Successors @ A1 @ Y3 )
=> ~ ( member8406446414694345712od_v_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_866_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,A1: set_o,A22: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4846409930415309433_set_o @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: set_o] :
( ( sCC_Bl4846409930415309433_set_o @ Successors @ A1 @ Y3 )
=> ~ ( member_set_o @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_867_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y3 )
=> ~ ( member_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_868_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_o,Successors: $o > set_o,A1: $o,A22: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl6452251072014827417_end_o @ Successors @ A1 @ A22 )
= ( ? [X2: $o] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: $o,Y: $o,Z5: $o] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl6452251072014827417_end_o @ Successors @ X2 @ Y )
& ( member_o @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_869_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y: product_prod_v_v,Z5: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y )
& ( member7453568604450474000od_v_v @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_870_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,A1: set_v,A22: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1820819216308814196_set_v @ Successors @ A1 @ A22 )
= ( ? [X2: set_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_v,Y: set_v,Z5: set_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl1820819216308814196_set_v @ Successors @ X2 @ Y )
& ( member_set_v @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_871_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,A1: set_Product_prod_v_v,A22: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3679296176305320008od_v_v @ Successors @ A1 @ A22 )
= ( ? [X2: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z5: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl3679296176305320008od_v_v @ Successors @ X2 @ Y )
& ( member8406446414694345712od_v_v @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_872_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,A1: set_o,A22: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4846409930415309433_set_o @ Successors @ A1 @ A22 )
= ( ? [X2: set_o] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: set_o,Y: set_o,Z5: set_o] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl4846409930415309433_set_o @ Successors @ X2 @ Y )
& ( member_set_o @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_873_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y )
& ( member_v @ Z5 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_874_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ X5 ) ) ).
% graph.re_refl
thf(fact_875_graph_Ore__succ,axiom,
! [Vertices: set_o,Successors: $o > set_o,X5: $o,Y2: $o,Z2: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl6452251072014827417_end_o @ Successors @ X5 @ Y2 )
=> ( ( member_o @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl6452251072014827417_end_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_876_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X5: product_prod_v_v,Y2: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X5 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_877_graph_Ore__succ,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,X5: set_v,Y2: set_v,Z2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1820819216308814196_set_v @ Successors @ X5 @ Y2 )
=> ( ( member_set_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl1820819216308814196_set_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_878_graph_Ore__succ,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3679296176305320008od_v_v @ Successors @ X5 @ Y2 )
=> ( ( member8406446414694345712od_v_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl3679296176305320008od_v_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_879_graph_Ore__succ,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,X5: set_o,Y2: set_o,Z2: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl4846409930415309433_set_o @ Successors @ X5 @ Y2 )
=> ( ( member_set_o @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl4846409930415309433_set_o @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_880_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X5: v,Y2: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ Y2 )
=> ( ( member_v @ Z2 @ ( Successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X5 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_881_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_882_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N3: v] :
( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_883_pigeonhole__infinite,axiom,
! [A: set_o,F: $o > v] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_v @ ( image_o_v @ F @ A ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_884_pigeonhole__infinite,axiom,
! [A: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ~ ( finite3348123685078250256od_v_v @ A )
=> ( ( finite_finite_v @ ( image_6152814753742948081_v_v_v @ F @ A ) )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A )
& ~ ( finite3348123685078250256od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [A4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_885_pigeonhole__infinite,axiom,
! [A: set_se8455005133513928103od_v_v,F: set_Product_prod_v_v > v] :
( ~ ( finite6084192165098772208od_v_v @ A )
=> ( ( finite_finite_v @ ( image_1718686574370184977_v_v_v @ F @ A ) )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
& ~ ( finite6084192165098772208od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [A4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_886_pigeonhole__infinite,axiom,
! [A: set_set_o,F: set_o > v] :
( ~ ( finite_finite_set_o @ A )
=> ( ( finite_finite_v @ ( image_set_o_v @ F @ A ) )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A )
& ~ ( finite_finite_set_o
@ ( collect_set_o
@ ^ [A4: set_o] :
( ( member_set_o @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_887_pigeonhole__infinite,axiom,
! [A: set_set_v,F: set_v > v] :
( ~ ( finite_finite_set_v @ A )
=> ( ( finite_finite_v @ ( image_set_v_v @ F @ A ) )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A )
& ~ ( finite_finite_set_v
@ ( collect_set_v
@ ^ [A4: set_v] :
( ( member_set_v @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_888_pigeonhole__infinite,axiom,
! [A: set_v,F: v > v] :
( ~ ( finite_finite_v @ A )
=> ( ( finite_finite_v @ ( image_v_v @ F @ A ) )
=> ? [X4: v] :
( ( member_v @ X4 @ A )
& ~ ( finite_finite_v
@ ( collect_v
@ ^ [A4: v] :
( ( member_v @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_889_finite__image__set2,axiom,
! [P: set_v > $o,Q: set_v > $o,F: set_v > set_v > set_v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_set_v @ ( collect_set_v @ Q ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_v,Y: set_v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_890_finite__image__set2,axiom,
! [P: set_v > $o,Q: set_v > $o,F: set_v > set_v > v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_set_v @ ( collect_set_v @ Q ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: set_v,Y: set_v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_891_finite__image__set2,axiom,
! [P: set_v > $o,Q: v > $o,F: set_v > v > set_v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_v @ ( collect_v @ Q ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_v,Y: v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_892_finite__image__set2,axiom,
! [P: set_v > $o,Q: v > $o,F: set_v > v > v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( ( finite_finite_v @ ( collect_v @ Q ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: set_v,Y: v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_893_finite__image__set2,axiom,
! [P: v > $o,Q: set_v > $o,F: v > set_v > set_v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_set_v @ ( collect_set_v @ Q ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v,Y: set_v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_894_finite__image__set2,axiom,
! [P: v > $o,Q: set_v > $o,F: v > set_v > v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_set_v @ ( collect_set_v @ Q ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: v,Y: set_v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_895_finite__image__set2,axiom,
! [P: v > $o,Q: v > $o,F: v > v > set_v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_v @ ( collect_v @ Q ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v,Y: v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_896_finite__image__set2,axiom,
! [P: v > $o,Q: v > $o,F: v > v > v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( ( finite_finite_v @ ( collect_v @ Q ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: v,Y: v] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_897_finite__image__set,axiom,
! [P: set_v > $o,F: set_v > set_v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: set_v] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_898_finite__image__set,axiom,
! [P: set_v > $o,F: set_v > v] :
( ( finite_finite_set_v @ ( collect_set_v @ P ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: set_v] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_899_finite__image__set,axiom,
! [P: v > $o,F: v > set_v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( finite_finite_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_900_finite__image__set,axiom,
! [P: v > $o,F: v > v] :
( ( finite_finite_v @ ( collect_v @ P ) )
=> ( finite_finite_v
@ ( collect_v
@ ^ [Uu: v] :
? [X2: v] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_901_le__cSup__finite,axiom,
! [X3: set_set_set_v,X5: set_set_v] :
( ( finite8701002811114149628_set_v @ X3 )
=> ( ( member_set_set_v @ X5 @ X3 )
=> ( ord_le5216385588623774835_set_v @ X5 @ ( comple5450237519782573632_set_v @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_902_le__cSup__finite,axiom,
! [X3: set_se2157405561750842759od_v_v,X5: set_se8455005133513928103od_v_v] :
( ( finite2944697185239447248od_v_v @ X3 )
=> ( ( member5511408251247217616od_v_v @ X5 @ X3 )
=> ( ord_le4714265922333009223od_v_v @ X5 @ ( comple6800410395631251988od_v_v @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_903_le__cSup__finite,axiom,
! [X3: set_set_v,X5: set_v] :
( ( finite_finite_set_v @ X3 )
=> ( ( member_set_v @ X5 @ X3 )
=> ( ord_less_eq_set_v @ X5 @ ( comple2307003700295860064_set_v @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_904_le__cSup__finite,axiom,
! [X3: set_o,X5: $o] :
( ( finite_finite_o @ X3 )
=> ( ( member_o @ X5 @ X3 )
=> ( ord_less_eq_o @ X5 @ ( complete_Sup_Sup_o @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_905_le__cSup__finite,axiom,
! [X3: set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ X3 )
=> ( ( member8406446414694345712od_v_v @ X5 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ ( comple5788137035815166516od_v_v @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_906_le__cSup__finite,axiom,
! [X3: set_set_o,X5: set_o] :
( ( finite_finite_set_o @ X3 )
=> ( ( member_set_o @ X5 @ X3 )
=> ( ord_less_eq_set_o @ X5 @ ( comple90263536869209701_set_o @ X3 ) ) ) ) ).
% le_cSup_finite
thf(fact_907_finite__surj,axiom,
! [A: set_v,B: set_v,F: v > v] :
( ( finite_finite_v @ A )
=> ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
=> ( finite_finite_v @ B ) ) ) ).
% finite_surj
thf(fact_908_finite__surj,axiom,
! [A: set_v,B: set_Product_prod_v_v,F: v > product_prod_v_v] :
( ( finite_finite_v @ A )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
=> ( finite3348123685078250256od_v_v @ B ) ) ) ).
% finite_surj
thf(fact_909_finite__surj,axiom,
! [A: set_v,B: set_o,F: v > $o] :
( ( finite_finite_v @ A )
=> ( ( ord_less_eq_set_o @ B @ ( image_v_o @ F @ A ) )
=> ( finite_finite_o @ B ) ) ) ).
% finite_surj
thf(fact_910_finite__surj,axiom,
! [A: set_v,B: set_set_v,F: v > set_v] :
( ( finite_finite_v @ A )
=> ( ( ord_le5216385588623774835_set_v @ B @ ( image_v_set_v @ F @ A ) )
=> ( finite_finite_set_v @ B ) ) ) ).
% finite_surj
thf(fact_911_finite__surj,axiom,
! [A: set_v,B: set_se8455005133513928103od_v_v,F: v > set_Product_prod_v_v] :
( ( finite_finite_v @ A )
=> ( ( ord_le4714265922333009223od_v_v @ B @ ( image_181480991005670265od_v_v @ F @ A ) )
=> ( finite6084192165098772208od_v_v @ B ) ) ) ).
% finite_surj
thf(fact_912_finite__subset__image,axiom,
! [B: set_v,F: v > v,A: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A ) )
=> ? [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
& ( finite_finite_v @ C3 )
& ( B
= ( image_v_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_913_finite__subset__image,axiom,
! [B: set_v,F: $o > v,A: set_o] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_o_v @ F @ A ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_914_finite__subset__image,axiom,
! [B: set_o,F: v > $o,A: set_v] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_v_o @ F @ A ) )
=> ? [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
& ( finite_finite_v @ C3 )
& ( B
= ( image_v_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_915_finite__subset__image,axiom,
! [B: set_o,F: $o > $o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_916_finite__subset__image,axiom,
! [B: set_v,F: set_v > v,A: set_set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_set_v_v @ F @ A ) )
=> ? [C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ A )
& ( finite_finite_set_v @ C3 )
& ( B
= ( image_set_v_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_917_finite__subset__image,axiom,
! [B: set_o,F: set_v > $o,A: set_set_v] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_set_v_o @ F @ A ) )
=> ? [C3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C3 @ A )
& ( finite_finite_set_v @ C3 )
& ( B
= ( image_set_v_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_918_finite__subset__image,axiom,
! [B: set_set_v,F: v > set_v,A: set_v] :
( ( finite_finite_set_v @ B )
=> ( ( ord_le5216385588623774835_set_v @ B @ ( image_v_set_v @ F @ A ) )
=> ? [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
& ( finite_finite_v @ C3 )
& ( B
= ( image_v_set_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_919_finite__subset__image,axiom,
! [B: set_set_v,F: $o > set_v,A: set_o] :
( ( finite_finite_set_v @ B )
=> ( ( ord_le5216385588623774835_set_v @ B @ ( image_o_set_v @ F @ A ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_set_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_920_finite__subset__image,axiom,
! [B: set_v,F: product_prod_v_v > v,A: set_Product_prod_v_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A ) )
=> ? [C3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C3 @ A )
& ( finite3348123685078250256od_v_v @ C3 )
& ( B
= ( image_6152814753742948081_v_v_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_921_finite__subset__image,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A: set_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A ) )
=> ? [C3: set_v] :
( ( ord_less_eq_set_v @ C3 @ A )
& ( finite_finite_v @ C3 )
& ( B
= ( image_9222788639401671577od_v_v @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_922_ex__finite__subset__image,axiom,
! [F: v > v,A: set_v,P: set_v > $o] :
( ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A )
& ( P @ ( image_v_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_923_ex__finite__subset__image,axiom,
! [F: $o > v,A: set_o,P: set_v > $o] :
( ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_o_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A )
& ( P @ ( image_o_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_924_ex__finite__subset__image,axiom,
! [F: v > $o,A: set_v,P: set_o > $o] :
( ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_v_o @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A )
& ( P @ ( image_v_o @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_925_ex__finite__subset__image,axiom,
! [F: $o > $o,A: set_o,P: set_o > $o] :
( ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A )
& ( P @ ( image_o_o @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_926_ex__finite__subset__image,axiom,
! [F: set_v > v,A: set_set_v,P: set_v > $o] :
( ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_set_v_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_v] :
( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ A )
& ( P @ ( image_set_v_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_927_ex__finite__subset__image,axiom,
! [F: set_v > $o,A: set_set_v,P: set_o > $o] :
( ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_set_v_o @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_v] :
( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ A )
& ( P @ ( image_set_v_o @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_928_ex__finite__subset__image,axiom,
! [F: v > set_v,A: set_v,P: set_set_v > $o] :
( ( ? [B2: set_set_v] :
( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ ( image_v_set_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A )
& ( P @ ( image_v_set_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_929_ex__finite__subset__image,axiom,
! [F: $o > set_v,A: set_o,P: set_set_v > $o] :
( ( ? [B2: set_set_v] :
( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ ( image_o_set_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A )
& ( P @ ( image_o_set_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_930_ex__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
( ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A )
& ( P @ ( image_6152814753742948081_v_v_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_931_ex__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
( ( ? [B2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_v] :
( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A )
& ( P @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_932_all__finite__subset__image,axiom,
! [F: v > v,A: set_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A ) )
=> ( P @ ( image_v_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_933_all__finite__subset__image,axiom,
! [F: $o > v,A: set_o,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_o_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A ) )
=> ( P @ ( image_o_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_934_all__finite__subset__image,axiom,
! [F: v > $o,A: set_v,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_v_o @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A ) )
=> ( P @ ( image_v_o @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_935_all__finite__subset__image,axiom,
! [F: $o > $o,A: set_o,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A ) )
=> ( P @ ( image_o_o @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_936_all__finite__subset__image,axiom,
! [F: set_v > v,A: set_set_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_set_v_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_v] :
( ( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ A ) )
=> ( P @ ( image_set_v_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_937_all__finite__subset__image,axiom,
! [F: set_v > $o,A: set_set_v,P: set_o > $o] :
( ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ ( image_set_v_o @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_v] :
( ( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ A ) )
=> ( P @ ( image_set_v_o @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_938_all__finite__subset__image,axiom,
! [F: v > set_v,A: set_v,P: set_set_v > $o] :
( ( ! [B2: set_set_v] :
( ( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ ( image_v_set_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A ) )
=> ( P @ ( image_v_set_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_939_all__finite__subset__image,axiom,
! [F: $o > set_v,A: set_o,P: set_set_v > $o] :
( ( ! [B2: set_set_v] :
( ( ( finite_finite_set_v @ B2 )
& ( ord_le5216385588623774835_set_v @ B2 @ ( image_o_set_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A ) )
=> ( P @ ( image_o_set_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_940_all__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A ) )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_941_all__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B2: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_v] :
( ( ( finite_finite_v @ B2 )
& ( ord_less_eq_set_v @ B2 @ A ) )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_942_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_943_graph_Ounite__S__equal,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,E: sCC_Bl337355980704484737t_unit,W: set_v,V: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl4608127289153132392t_unit @ Successors @ E )
=> ( ( member_set_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl1947314421851746340t_unit @ E @ V ) )
=> ( ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl8056216847179350138t_unit @ E ) )
=> ( ( comple5450237519782573632_set_v
@ ( collect_set_set_v
@ ^ [Uu: set_set_v] :
? [N2: set_v] :
( ( Uu
= ( sCC_Bl4451002444308059158t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) @ N2 ) )
& ( member_set_v @ N2 @ ( set_set_v2 @ ( sCC_Bl5937180991433647859t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple5450237519782573632_set_v
@ ( collect_set_set_v
@ ^ [Uu: set_set_v] :
? [N2: set_v] :
( ( Uu
= ( sCC_Bl4451002444308059158t_unit @ E @ N2 ) )
& ( member_set_v @ N2 @ ( set_set_v2 @ ( sCC_Bl5937180991433647859t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_944_graph_Ounite__S__equal,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,E: sCC_Bl471817294227068333t_unit,W: set_Product_prod_v_v,V: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6965693132275972500t_unit @ Successors @ E )
=> ( ( member8406446414694345712od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl454723783576663632t_unit @ E @ V ) )
=> ( ( member8406446414694345712od_v_v @ W @ ( sCC_Bl647639559028580801t_unit @ E ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl2694530823580680358t_unit @ E ) )
=> ( ( comple6800410395631251988od_v_v
@ ( collec5787501045982241298od_v_v
@ ^ [Uu: set_se8455005133513928103od_v_v] :
? [N2: set_Product_prod_v_v] :
( ( Uu
= ( sCC_Bl384457957625567298t_unit @ ( sCC_Bl2743269859033768757od_v_v @ V @ W @ E ) @ N2 ) )
& ( member8406446414694345712od_v_v @ N2 @ ( set_se5775211966769592924od_v_v @ ( sCC_Bl7995652726447031839t_unit @ ( sCC_Bl2743269859033768757od_v_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple6800410395631251988od_v_v
@ ( collec5787501045982241298od_v_v
@ ^ [Uu: set_se8455005133513928103od_v_v] :
? [N2: set_Product_prod_v_v] :
( ( Uu
= ( sCC_Bl384457957625567298t_unit @ E @ N2 ) )
& ( member8406446414694345712od_v_v @ N2 @ ( set_se5775211966769592924od_v_v @ ( sCC_Bl7995652726447031839t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_945_graph_Ounite__S__equal,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,E: sCC_Bl8746163187549392316t_unit,W: set_o,V: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl7435977579827688099t_unit @ Successors @ E )
=> ( ( member_set_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl5177287912846518623t_unit @ E @ V ) )
=> ( ( member_set_o @ W @ ( sCC_Bl8197997162987607632t_unit @ E ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl4929516568782881973t_unit @ E ) )
=> ( ( comple4436988014476444997_set_o
@ ( collect_set_set_o
@ ^ [Uu: set_set_o] :
? [N2: set_o] :
( ( Uu
= ( sCC_Bl5430157894217623121t_unit @ ( sCC_Bl5184775513941676262_set_o @ V @ W @ E ) @ N2 ) )
& ( member_set_o @ N2 @ ( set_set_o2 @ ( sCC_Bl1631577087933863854t_unit @ ( sCC_Bl5184775513941676262_set_o @ V @ W @ E ) ) ) ) ) ) )
= ( comple4436988014476444997_set_o
@ ( collect_set_set_o
@ ^ [Uu: set_set_o] :
? [N2: set_o] :
( ( Uu
= ( sCC_Bl5430157894217623121t_unit @ E @ N2 ) )
& ( member_set_o @ N2 @ ( set_set_o2 @ ( sCC_Bl1631577087933863854t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_946_graph_Ounite__S__equal,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N2: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N2 ) )
& ( member7453568604450474000od_v_v @ N2 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N2: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ E @ N2 ) )
& ( member7453568604450474000od_v_v @ N2 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_947_graph_Ounite__S__equal,axiom,
! [Vertices: set_o,Successors: $o > set_o,E: sCC_Bl3933283745082679452t_unit,W: $o,V: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl5436259119265389955t_unit @ Successors @ E )
=> ( ( member_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl2127731265848312895t_unit @ E @ V ) )
=> ( ( member_o @ W @ ( sCC_Bl3035372653189477680t_unit @ E ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl4316552501791471509t_unit @ E ) )
=> ( ( comple90263536869209701_set_o
@ ( collect_set_o
@ ^ [Uu: set_o] :
? [N2: $o] :
( ( Uu
= ( sCC_Bl7367474962761402673t_unit @ ( sCC_Bloemen_unite_o @ V @ W @ E ) @ N2 ) )
& ( member_o @ N2 @ ( set_o2 @ ( sCC_Bl2148746266190622350t_unit @ ( sCC_Bloemen_unite_o @ V @ W @ E ) ) ) ) ) ) )
= ( comple90263536869209701_set_o
@ ( collect_set_o
@ ^ [Uu: set_o] :
? [N2: $o] :
( ( Uu
= ( sCC_Bl7367474962761402673t_unit @ E @ N2 ) )
& ( member_o @ N2 @ ( set_o2 @ ( sCC_Bl2148746266190622350t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_948_graph_Ounite__S__equal,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) )
& ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_949_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_950_Fpow__mono,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ord_le5216385588623774835_set_v @ ( finite_Fpow_v @ A ) @ ( finite_Fpow_v @ B ) ) ) ).
% Fpow_mono
thf(fact_951_Fpow__mono,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ord_le4714265922333009223od_v_v @ ( finite315275465967970893od_v_v @ A ) @ ( finite315275465967970893od_v_v @ B ) ) ) ).
% Fpow_mono
thf(fact_952_Fpow__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_le4374716579403074808_set_o @ ( finite_Fpow_o @ A ) @ ( finite_Fpow_o @ B ) ) ) ).
% Fpow_mono
thf(fact_953_Fpow__mono,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ord_le8117609702905084755_set_v @ ( finite_Fpow_set_v @ A ) @ ( finite_Fpow_set_v @ B ) ) ) ).
% Fpow_mono
thf(fact_954_Fpow__mono,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ord_le4688432828652561703od_v_v @ ( finite1064079062865762861od_v_v @ A ) @ ( finite1064079062865762861od_v_v @ B ) ) ) ).
% Fpow_mono
thf(fact_955_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y: v] :
( ( member_v @ Y @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y ) ) ) ) ) ).
% is_subscc_def
thf(fact_956_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_957_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_958_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E2 )
= ( sCC_Bl1090238580953940555t_unit @ E ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_959_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_960_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_961_pfx_I1_J,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% pfx(1)
thf(fact_962_finite__subset__Union,axiom,
! [A: set_set_v,B6: set_set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( ord_le5216385588623774835_set_v @ A @ ( comple5450237519782573632_set_v @ B6 ) )
=> ~ ! [F4: set_set_set_v] :
( ( finite8701002811114149628_set_v @ F4 )
=> ( ( ord_le8117609702905084755_set_v @ F4 @ B6 )
=> ~ ( ord_le5216385588623774835_set_v @ A @ ( comple5450237519782573632_set_v @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_963_finite__subset__Union,axiom,
! [A: set_se8455005133513928103od_v_v,B6: set_se2157405561750842759od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( ord_le4714265922333009223od_v_v @ A @ ( comple6800410395631251988od_v_v @ B6 ) )
=> ~ ! [F4: set_se2157405561750842759od_v_v] :
( ( finite2944697185239447248od_v_v @ F4 )
=> ( ( ord_le4688432828652561703od_v_v @ F4 @ B6 )
=> ~ ( ord_le4714265922333009223od_v_v @ A @ ( comple6800410395631251988od_v_v @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_964_finite__subset__Union,axiom,
! [A: set_v,B6: set_set_v] :
( ( finite_finite_v @ A )
=> ( ( ord_less_eq_set_v @ A @ ( comple2307003700295860064_set_v @ B6 ) )
=> ~ ! [F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ( ord_le5216385588623774835_set_v @ F4 @ B6 )
=> ~ ( ord_less_eq_set_v @ A @ ( comple2307003700295860064_set_v @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_965_finite__subset__Union,axiom,
! [A: set_Product_prod_v_v,B6: set_se8455005133513928103od_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ ( comple5788137035815166516od_v_v @ B6 ) )
=> ~ ! [F4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ F4 )
=> ( ( ord_le4714265922333009223od_v_v @ F4 @ B6 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ ( comple5788137035815166516od_v_v @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_966_finite__subset__Union,axiom,
! [A: set_o,B6: set_set_o] :
( ( finite_finite_o @ A )
=> ( ( ord_less_eq_set_o @ A @ ( comple90263536869209701_set_o @ B6 ) )
=> ~ ! [F4: set_set_o] :
( ( finite_finite_set_o @ F4 )
=> ( ( ord_le4374716579403074808_set_o @ F4 @ B6 )
=> ~ ( ord_less_eq_set_o @ A @ ( comple90263536869209701_set_o @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_967_DiffI,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ A )
=> ( ~ ( member_v @ C2 @ B )
=> ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_968_DiffI,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ A )
=> ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_969_DiffI,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_970_DiffI,axiom,
! [C2: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ A )
=> ( ~ ( member_set_v @ C2 @ B )
=> ( member_set_v @ C2 @ ( minus_7228012346218142266_set_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_971_DiffI,axiom,
! [C2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ C2 @ A )
=> ( ~ ( member8406446414694345712od_v_v @ C2 @ B )
=> ( member8406446414694345712od_v_v @ C2 @ ( minus_7679383599658060814od_v_v @ A @ B ) ) ) ) ).
% DiffI
thf(fact_972_DiffI,axiom,
! [C2: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ A )
=> ( ~ ( member_set_o @ C2 @ B )
=> ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_973_Diff__iff,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
= ( ( member_v @ C2 @ A )
& ~ ( member_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_974_Diff__iff,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
= ( ( member_o @ C2 @ A )
& ~ ( member_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_975_Diff__iff,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A )
& ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_976_Diff__iff,axiom,
! [C2: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( minus_7228012346218142266_set_v @ A @ B ) )
= ( ( member_set_v @ C2 @ A )
& ~ ( member_set_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_977_Diff__iff,axiom,
! [C2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ C2 @ ( minus_7679383599658060814od_v_v @ A @ B ) )
= ( ( member8406446414694345712od_v_v @ C2 @ A )
& ~ ( member8406446414694345712od_v_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_978_Diff__iff,axiom,
! [C2: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A @ B ) )
= ( ( member_set_o @ C2 @ A )
& ~ ( member_set_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_979_finite__Diff,axiom,
! [A: set_v,B: set_v] :
( ( finite_finite_v @ A )
=> ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) ) ) ).
% finite_Diff
thf(fact_980_finite__Diff2,axiom,
! [B: set_v,A: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A @ B ) )
= ( finite_finite_v @ A ) ) ) ).
% finite_Diff2
thf(fact_981_list_Oset__sel_I2_J,axiom,
! [A2: list_o,X5: $o] :
( ( A2 != nil_o )
=> ( ( member_o @ X5 @ ( set_o2 @ ( tl_o @ A2 ) ) )
=> ( member_o @ X5 @ ( set_o2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_982_list_Oset__sel_I2_J,axiom,
! [A2: list_P7986770385144383213od_v_v,X5: product_prod_v_v] :
( ( A2 != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X5 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A2 ) ) )
=> ( member7453568604450474000od_v_v @ X5 @ ( set_Product_prod_v_v2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_983_list_Oset__sel_I2_J,axiom,
! [A2: list_set_v,X5: set_v] :
( ( A2 != nil_set_v )
=> ( ( member_set_v @ X5 @ ( set_set_v2 @ ( tl_set_v @ A2 ) ) )
=> ( member_set_v @ X5 @ ( set_set_v2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_984_list_Oset__sel_I2_J,axiom,
! [A2: list_s2556036278619108045od_v_v,X5: set_Product_prod_v_v] :
( ( A2 != nil_se4518395726187269069od_v_v )
=> ( ( member8406446414694345712od_v_v @ X5 @ ( set_se5775211966769592924od_v_v @ ( tl_set8448285760204898230od_v_v @ A2 ) ) )
=> ( member8406446414694345712od_v_v @ X5 @ ( set_se5775211966769592924od_v_v @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_985_list_Oset__sel_I2_J,axiom,
! [A2: list_set_o,X5: set_o] :
( ( A2 != nil_set_o )
=> ( ( member_set_o @ X5 @ ( set_set_o2 @ ( tl_set_o @ A2 ) ) )
=> ( member_set_o @ X5 @ ( set_set_o2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_986_list_Oset__sel_I2_J,axiom,
! [A2: list_v,X5: v] :
( ( A2 != nil_v )
=> ( ( member_v @ X5 @ ( set_v2 @ ( tl_v @ A2 ) ) )
=> ( member_v @ X5 @ ( set_v2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_987_DiffE,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( ( member_v @ C2 @ A )
=> ( member_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_988_DiffE,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% DiffE
thf(fact_989_DiffE,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_990_DiffE,axiom,
! [C2: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( minus_7228012346218142266_set_v @ A @ B ) )
=> ~ ( ( member_set_v @ C2 @ A )
=> ( member_set_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_991_DiffE,axiom,
! [C2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ C2 @ ( minus_7679383599658060814od_v_v @ A @ B ) )
=> ~ ( ( member8406446414694345712od_v_v @ C2 @ A )
=> ( member8406446414694345712od_v_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_992_DiffE,axiom,
! [C2: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A @ B ) )
=> ~ ( ( member_set_o @ C2 @ A )
=> ( member_set_o @ C2 @ B ) ) ) ).
% DiffE
thf(fact_993_DiffD1,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ( member_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_994_DiffD1,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ( member_o @ C2 @ A ) ) ).
% DiffD1
thf(fact_995_DiffD1,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ( member7453568604450474000od_v_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_996_DiffD1,axiom,
! [C2: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( minus_7228012346218142266_set_v @ A @ B ) )
=> ( member_set_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_997_DiffD1,axiom,
! [C2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ C2 @ ( minus_7679383599658060814od_v_v @ A @ B ) )
=> ( member8406446414694345712od_v_v @ C2 @ A ) ) ).
% DiffD1
thf(fact_998_DiffD1,axiom,
! [C2: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A @ B ) )
=> ( member_set_o @ C2 @ A ) ) ).
% DiffD1
thf(fact_999_DiffD2,axiom,
! [C2: v,A: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A @ B ) )
=> ~ ( member_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_1000_DiffD2,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( member_o @ C2 @ B ) ) ).
% DiffD2
thf(fact_1001_DiffD2,axiom,
! [C2: product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_1002_DiffD2,axiom,
! [C2: set_v,A: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( minus_7228012346218142266_set_v @ A @ B ) )
=> ~ ( member_set_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_1003_DiffD2,axiom,
! [C2: set_Product_prod_v_v,A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ C2 @ ( minus_7679383599658060814od_v_v @ A @ B ) )
=> ~ ( member8406446414694345712od_v_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_1004_DiffD2,axiom,
! [C2: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C2 @ ( minus_4899875422681990719_set_o @ A @ B ) )
=> ~ ( member_set_o @ C2 @ B ) ) ).
% DiffD2
thf(fact_1005_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_1006_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_1007_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A3 )
& ~ ( member_v @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1008_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B2: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ~ ( member_o @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1009_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ~ ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1010_set__diff__eq,axiom,
( minus_7679383599658060814od_v_v
= ( ^ [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( collec8263177866097347122od_v_v
@ ^ [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
& ~ ( member8406446414694345712od_v_v @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1011_set__diff__eq,axiom,
( minus_4899875422681990719_set_o
= ( ^ [A3: set_set_o,B2: set_set_o] :
( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A3 )
& ~ ( member_set_o @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1012_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
& ~ ( member_set_v @ X2 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1013_Diff__infinite__finite,axiom,
! [T2: set_v,S: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1014_Diff__mono,axiom,
! [A: set_v,C: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ ( minus_minus_set_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1015_Diff__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ ( minus_4183494784930505774od_v_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1016_Diff__mono,axiom,
! [A: set_o,C: set_o,D: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ C )
=> ( ( ord_less_eq_set_o @ D @ B )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ B ) @ ( minus_minus_set_o @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1017_Diff__mono,axiom,
! [A: set_set_v,C: set_set_v,D: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C )
=> ( ( ord_le5216385588623774835_set_v @ D @ B )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ ( minus_7228012346218142266_set_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1018_Diff__mono,axiom,
! [A: set_se8455005133513928103od_v_v,C: set_se8455005133513928103od_v_v,D: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ C )
=> ( ( ord_le4714265922333009223od_v_v @ D @ B )
=> ( ord_le4714265922333009223od_v_v @ ( minus_7679383599658060814od_v_v @ A @ B ) @ ( minus_7679383599658060814od_v_v @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1019_Diff__subset,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1020_Diff__subset,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1021_Diff__subset,axiom,
! [A: set_o,B: set_o] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1022_Diff__subset,axiom,
! [A: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1023_Diff__subset,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ ( minus_7679383599658060814od_v_v @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1024_double__diff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1025_double__diff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1026_double__diff,axiom,
! [A: set_o,B: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C )
=> ( ( minus_minus_set_o @ B @ ( minus_minus_set_o @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1027_double__diff,axiom,
! [A: set_set_v,B: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( ord_le5216385588623774835_set_v @ B @ C )
=> ( ( minus_7228012346218142266_set_v @ B @ ( minus_7228012346218142266_set_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1028_double__diff,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v,C: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ B )
=> ( ( ord_le4714265922333009223od_v_v @ B @ C )
=> ( ( minus_7679383599658060814od_v_v @ B @ ( minus_7679383599658060814od_v_v @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1029_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y: v] :
( ( member_v @ Y @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_1030_graph_OsccE,axiom,
! [Vertices: set_o,Successors: $o > set_o,S: set_o,X5: $o,Y2: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_o @ Successors @ S )
=> ( ( member_o @ X5 @ S )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ Y2 @ X5 )
=> ( member_o @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1031_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X5: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X5 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X5 )
=> ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1032_graph_OsccE,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X5: set_v,Y2: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
=> ( ( member_set_v @ X5 @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y2 @ X5 )
=> ( member_set_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1033_graph_OsccE,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,S: set_se8455005133513928103od_v_v,X5: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl1365425272170373485od_v_v @ Successors @ S )
=> ( ( member8406446414694345712od_v_v @ X5 @ S )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl8444643370470080105od_v_v @ Successors @ Y2 @ X5 )
=> ( member8406446414694345712od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1034_graph_OsccE,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,S: set_set_o,X5: set_o,Y2: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl7321494450242560798_set_o @ Successors @ S )
=> ( ( member_set_o @ X5 @ S )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl4440840191236725274_set_o @ Successors @ Y2 @ X5 )
=> ( member_set_o @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1035_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X5: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X5 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X5 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X5 )
=> ( member_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1036_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_1037_finite__list,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ? [Xs2: list_v] :
( ( set_v2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_1038_subset__code_I1_J,axiom,
! [Xs: list_set_o,B: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( set_set_o2 @ Xs ) @ B )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ ( set_set_o2 @ Xs ) )
=> ( member_set_o @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1039_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1040_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1041_subset__code_I1_J,axiom,
! [Xs: list_o,B: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B )
= ( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( member_o @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1042_subset__code_I1_J,axiom,
! [Xs: list_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( set_set_v2 @ Xs ) @ B )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ ( set_set_v2 @ Xs ) )
=> ( member_set_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1043_subset__code_I1_J,axiom,
! [Xs: list_s2556036278619108045od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ ( set_se5775211966769592924od_v_v @ Xs ) @ B )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ ( set_se5775211966769592924od_v_v @ Xs ) )
=> ( member8406446414694345712od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1044_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_1045_graph_Ounite__S__tl,axiom,
! [Vertices: set_o,Successors: $o > set_o,E: sCC_Bl3933283745082679452t_unit,W: $o,V: $o,N: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl5436259119265389955t_unit @ Successors @ E )
=> ( ( member_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl2127731265848312895t_unit @ E @ V ) )
=> ( ( member_o @ W @ ( sCC_Bl3035372653189477680t_unit @ E ) )
=> ( ~ ( member_o @ W @ ( sCC_Bl4316552501791471509t_unit @ E ) )
=> ( ( member_o @ N @ ( set_o2 @ ( tl_o @ ( sCC_Bl2148746266190622350t_unit @ ( sCC_Bloemen_unite_o @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl7367474962761402673t_unit @ ( sCC_Bloemen_unite_o @ V @ W @ E ) @ N )
= ( sCC_Bl7367474962761402673t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1046_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1047_graph_Ounite__S__tl,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,E: sCC_Bl337355980704484737t_unit,W: set_v,V: set_v,N: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl4608127289153132392t_unit @ Successors @ E )
=> ( ( member_set_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl1947314421851746340t_unit @ E @ V ) )
=> ( ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl8056216847179350138t_unit @ E ) )
=> ( ( member_set_v @ N @ ( set_set_v2 @ ( tl_set_v @ ( sCC_Bl5937180991433647859t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl4451002444308059158t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) @ N )
= ( sCC_Bl4451002444308059158t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1048_graph_Ounite__S__tl,axiom,
! [Vertices: set_se8455005133513928103od_v_v,Successors: set_Product_prod_v_v > set_se8455005133513928103od_v_v,E: sCC_Bl471817294227068333t_unit,W: set_Product_prod_v_v,V: set_Product_prod_v_v,N: set_Product_prod_v_v] :
( ( sCC_Bl8431766404916698662od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6965693132275972500t_unit @ Successors @ E )
=> ( ( member8406446414694345712od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl454723783576663632t_unit @ E @ V ) )
=> ( ( member8406446414694345712od_v_v @ W @ ( sCC_Bl647639559028580801t_unit @ E ) )
=> ( ~ ( member8406446414694345712od_v_v @ W @ ( sCC_Bl2694530823580680358t_unit @ E ) )
=> ( ( member8406446414694345712od_v_v @ N @ ( set_se5775211966769592924od_v_v @ ( tl_set8448285760204898230od_v_v @ ( sCC_Bl7995652726447031839t_unit @ ( sCC_Bl2743269859033768757od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl384457957625567298t_unit @ ( sCC_Bl2743269859033768757od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl384457957625567298t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1049_graph_Ounite__S__tl,axiom,
! [Vertices: set_set_o,Successors: set_o > set_set_o,E: sCC_Bl8746163187549392316t_unit,W: set_o,V: set_o,N: set_o] :
( ( sCC_Bl7236177227677027671_set_o @ Vertices @ Successors )
=> ( ( sCC_Bl7435977579827688099t_unit @ Successors @ E )
=> ( ( member_set_o @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl5177287912846518623t_unit @ E @ V ) )
=> ( ( member_set_o @ W @ ( sCC_Bl8197997162987607632t_unit @ E ) )
=> ( ~ ( member_set_o @ W @ ( sCC_Bl4929516568782881973t_unit @ E ) )
=> ( ( member_set_o @ N @ ( set_set_o2 @ ( tl_set_o @ ( sCC_Bl1631577087933863854t_unit @ ( sCC_Bl5184775513941676262_set_o @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl5430157894217623121t_unit @ ( sCC_Bl5184775513941676262_set_o @ V @ W @ E ) @ N )
= ( sCC_Bl5430157894217623121t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1050_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1051_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_1052_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_1053_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_1054_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_1055_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
=> ( S3 = S ) ) ) ) ).
% is_scc_def
thf(fact_1056_ra__empty,axiom,
! [X5: v,Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 ) ) ).
% ra_empty
thf(fact_1057_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_1058_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1059_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1060_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_1061_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1062_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1063_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_1064_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_1065_all__not__in__conv,axiom,
! [A: set_set_v] :
( ( ! [X2: set_v] :
~ ( member_set_v @ X2 @ A ) )
= ( A = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_1066_all__not__in__conv,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( ! [X2: set_Product_prod_v_v] :
~ ( member8406446414694345712od_v_v @ X2 @ A ) )
= ( A = bot_bo3497076220358800403od_v_v ) ) ).
% all_not_in_conv
thf(fact_1067_all__not__in__conv,axiom,
! [A: set_set_o] :
( ( ! [X2: set_o] :
~ ( member_set_o @ X2 @ A ) )
= ( A = bot_bot_set_set_o ) ) ).
% all_not_in_conv
thf(fact_1068_all__not__in__conv,axiom,
! [A: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A ) )
= ( A = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_1069_all__not__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_1070_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_1071_empty__iff,axiom,
! [C2: set_v] :
~ ( member_set_v @ C2 @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_1072_empty__iff,axiom,
! [C2: set_Product_prod_v_v] :
~ ( member8406446414694345712od_v_v @ C2 @ bot_bo3497076220358800403od_v_v ) ).
% empty_iff
thf(fact_1073_empty__iff,axiom,
! [C2: set_o] :
~ ( member_set_o @ C2 @ bot_bot_set_set_o ) ).
% empty_iff
thf(fact_1074_empty__iff,axiom,
! [C2: v] :
~ ( member_v @ C2 @ bot_bot_set_v ) ).
% empty_iff
thf(fact_1075_empty__iff,axiom,
! [C2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C2 @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_1076_image__is__empty,axiom,
! [F: v > v,A: set_v] :
( ( ( image_v_v @ F @ A )
= bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1077_image__is__empty,axiom,
! [F: product_prod_v_v > v,A: set_Product_prod_v_v] :
( ( ( image_6152814753742948081_v_v_v @ F @ A )
= bot_bot_set_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1078_image__is__empty,axiom,
! [F: v > product_prod_v_v,A: set_v] :
( ( ( image_9222788639401671577od_v_v @ F @ A )
= bot_bo723834152578015283od_v_v )
= ( A = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1079_image__is__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
( ( ( image_781944334261467077od_v_v @ F @ A )
= bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1080_empty__is__image,axiom,
! [F: v > v,A: set_v] :
( ( bot_bot_set_v
= ( image_v_v @ F @ A ) )
= ( A = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1081_empty__is__image,axiom,
! [F: product_prod_v_v > v,A: set_Product_prod_v_v] :
( ( bot_bot_set_v
= ( image_6152814753742948081_v_v_v @ F @ A ) )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1082_empty__is__image,axiom,
! [F: v > product_prod_v_v,A: set_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_9222788639401671577od_v_v @ F @ A ) )
= ( A = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1083_empty__is__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_781944334261467077od_v_v @ F @ A ) )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1084_image__empty,axiom,
! [F: v > v] :
( ( image_v_v @ F @ bot_bot_set_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1085_image__empty,axiom,
! [F: v > product_prod_v_v] :
( ( image_9222788639401671577od_v_v @ F @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1086_image__empty,axiom,
! [F: product_prod_v_v > v] :
( ( image_6152814753742948081_v_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1087_image__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1088_empty__subsetI,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% empty_subsetI
thf(fact_1089_empty__subsetI,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% empty_subsetI
thf(fact_1090_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_1091_empty__subsetI,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% empty_subsetI
thf(fact_1092_empty__subsetI,axiom,
! [A: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ bot_bo3497076220358800403od_v_v @ A ) ).
% empty_subsetI
thf(fact_1093_subset__empty,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_1094_subset__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_1095_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_1096_subset__empty,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_1097_subset__empty,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A @ bot_bo3497076220358800403od_v_v )
= ( A = bot_bo3497076220358800403od_v_v ) ) ).
% subset_empty
thf(fact_1098_Sup__bot__conv_I1_J,axiom,
! [A: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1099_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1100_Sup__bot__conv_I1_J,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1101_Sup__bot__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1102_Sup__bot__conv_I2_J,axiom,
! [A: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1103_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1104_Sup__bot__conv_I2_J,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1105_Sup__bot__conv_I2_J,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1106_Diff__cancel,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ A )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_1107_Diff__cancel,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ A )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_1108_empty__Diff,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_1109_empty__Diff,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_1110_Diff__empty,axiom,
! [A: set_v] :
( ( minus_minus_set_v @ A @ bot_bot_set_v )
= A ) ).
% Diff_empty
thf(fact_1111_Diff__empty,axiom,
! [A: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% Diff_empty
thf(fact_1112_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_1113_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_1114_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_1115_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_1116_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_1117_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_1118_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_1119_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_1120_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_1121_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_1122_Diff__eq__empty__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1123_Diff__eq__empty__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1124_Diff__eq__empty__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1125_Diff__eq__empty__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1126_Diff__eq__empty__iff,axiom,
! [A: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ( minus_7679383599658060814od_v_v @ A @ B )
= bot_bo3497076220358800403od_v_v )
= ( ord_le4714265922333009223od_v_v @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1127_SUP__const,axiom,
! [A: set_v,F: set_v] :
( ( A != bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [I: v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1128_SUP__const,axiom,
! [A: set_Product_prod_v_v,F: set_v] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [I: product_prod_v_v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1129_SUP__const,axiom,
! [A: set_v,F: $o] :
( ( A != bot_bot_set_v )
=> ( ( complete_Sup_Sup_o
@ ( image_v_o
@ ^ [I: v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1130_SUP__const,axiom,
! [A: set_Product_prod_v_v,F: $o] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( complete_Sup_Sup_o
@ ( image_4100611349647363702_v_v_o
@ ^ [I: product_prod_v_v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1131_SUP__const,axiom,
! [A: set_v,F: set_Product_prod_v_v] :
( ( A != bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [I: v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1132_SUP__const,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [I: product_prod_v_v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1133_SUP__const,axiom,
! [A: set_v,F: set_o] :
( ( A != bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [I: v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1134_SUP__const,axiom,
! [A: set_Product_prod_v_v,F: set_o] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [I: product_prod_v_v] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1135_cSUP__const,axiom,
! [A: set_v,C2: set_v] :
( ( A != bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [X2: v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1136_cSUP__const,axiom,
! [A: set_Product_prod_v_v,C2: set_v] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [X2: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1137_cSUP__const,axiom,
! [A: set_v,C2: $o] :
( ( A != bot_bot_set_v )
=> ( ( complete_Sup_Sup_o
@ ( image_v_o
@ ^ [X2: v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1138_cSUP__const,axiom,
! [A: set_Product_prod_v_v,C2: $o] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( complete_Sup_Sup_o
@ ( image_4100611349647363702_v_v_o
@ ^ [X2: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1139_cSUP__const,axiom,
! [A: set_v,C2: set_Product_prod_v_v] :
( ( A != bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [X2: v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1140_cSUP__const,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [X2: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1141_cSUP__const,axiom,
! [A: set_v,C2: set_o] :
( ( A != bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [X2: v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1142_cSUP__const,axiom,
! [A: set_Product_prod_v_v,C2: set_o] :
( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [X2: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1143_UN__constant,axiom,
! [A: set_v,C2: set_v] :
( ( ( A = bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [Y: v] : C2
@ A ) )
= bot_bot_set_v ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1144_UN__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_v] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bot_set_v ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1145_UN__constant,axiom,
! [A: set_v,C2: set_Product_prod_v_v] :
( ( ( A = bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [Y: v] : C2
@ A ) )
= bot_bo723834152578015283od_v_v ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1146_UN__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bo723834152578015283od_v_v ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1147_UN__constant,axiom,
! [A: set_v,C2: set_o] :
( ( ( A = bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [Y: v] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1148_UN__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_o] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1149_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_1150_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_1151_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_1152_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X2: set_v] : $false ) ) ).
% empty_def
thf(fact_1153_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X2: v] : $false ) ) ).
% empty_def
thf(fact_1154_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_1155_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_1156_ex__in__conv,axiom,
! [A: set_set_v] :
( ( ? [X2: set_v] : ( member_set_v @ X2 @ A ) )
= ( A != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_1157_ex__in__conv,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( ? [X2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ X2 @ A ) )
= ( A != bot_bo3497076220358800403od_v_v ) ) ).
% ex_in_conv
thf(fact_1158_ex__in__conv,axiom,
! [A: set_set_o] :
( ( ? [X2: set_o] : ( member_set_o @ X2 @ A ) )
= ( A != bot_bot_set_set_o ) ) ).
% ex_in_conv
thf(fact_1159_ex__in__conv,axiom,
! [A: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A ) )
= ( A != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_1160_ex__in__conv,axiom,
! [A: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A ) )
= ( A != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_1161_equals0I,axiom,
! [A: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_1162_equals0I,axiom,
! [A: set_set_v] :
( ! [Y3: set_v] :
~ ( member_set_v @ Y3 @ A )
=> ( A = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_1163_equals0I,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ! [Y3: set_Product_prod_v_v] :
~ ( member8406446414694345712od_v_v @ Y3 @ A )
=> ( A = bot_bo3497076220358800403od_v_v ) ) ).
% equals0I
thf(fact_1164_equals0I,axiom,
! [A: set_set_o] :
( ! [Y3: set_o] :
~ ( member_set_o @ Y3 @ A )
=> ( A = bot_bot_set_set_o ) ) ).
% equals0I
thf(fact_1165_equals0I,axiom,
! [A: set_v] :
( ! [Y3: v] :
~ ( member_v @ Y3 @ A )
=> ( A = bot_bot_set_v ) ) ).
% equals0I
thf(fact_1166_equals0I,axiom,
! [A: set_Product_prod_v_v] :
( ! [Y3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y3 @ A )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_1167_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_1168_equals0D,axiom,
! [A: set_set_v,A2: set_v] :
( ( A = bot_bot_set_set_v )
=> ~ ( member_set_v @ A2 @ A ) ) ).
% equals0D
thf(fact_1169_equals0D,axiom,
! [A: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( A = bot_bo3497076220358800403od_v_v )
=> ~ ( member8406446414694345712od_v_v @ A2 @ A ) ) ).
% equals0D
thf(fact_1170_equals0D,axiom,
! [A: set_set_o,A2: set_o] :
( ( A = bot_bot_set_set_o )
=> ~ ( member_set_o @ A2 @ A ) ) ).
% equals0D
thf(fact_1171_equals0D,axiom,
! [A: set_v,A2: v] :
( ( A = bot_bot_set_v )
=> ~ ( member_v @ A2 @ A ) ) ).
% equals0D
thf(fact_1172_equals0D,axiom,
! [A: set_Product_prod_v_v,A2: product_prod_v_v] :
( ( A = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A2 @ A ) ) ).
% equals0D
thf(fact_1173_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_1174_emptyE,axiom,
! [A2: set_v] :
~ ( member_set_v @ A2 @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_1175_emptyE,axiom,
! [A2: set_Product_prod_v_v] :
~ ( member8406446414694345712od_v_v @ A2 @ bot_bo3497076220358800403od_v_v ) ).
% emptyE
thf(fact_1176_emptyE,axiom,
! [A2: set_o] :
~ ( member_set_o @ A2 @ bot_bot_set_set_o ) ).
% emptyE
thf(fact_1177_emptyE,axiom,
! [A2: v] :
~ ( member_v @ A2 @ bot_bot_set_v ) ).
% emptyE
thf(fact_1178_emptyE,axiom,
! [A2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A2 @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_1179_bot_Oextremum__uniqueI,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
=> ( A2 = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_1180_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_1181_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1182_bot_Oextremum__uniqueI,axiom,
! [A2: $o] :
( ( ord_less_eq_o @ A2 @ bot_bot_o )
=> ( A2 = bot_bot_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1183_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ bot_bot_set_set_v )
=> ( A2 = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_1184_bot_Oextremum__uniqueI,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ bot_bo3497076220358800403od_v_v )
=> ( A2 = bot_bo3497076220358800403od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_1185_bot_Oextremum__unique,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_1186_bot_Oextremum__unique,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_1187_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_1188_bot_Oextremum__unique,axiom,
! [A2: $o] :
( ( ord_less_eq_o @ A2 @ bot_bot_o )
= ( A2 = bot_bot_o ) ) ).
% bot.extremum_unique
thf(fact_1189_bot_Oextremum__unique,axiom,
! [A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ bot_bot_set_set_v )
= ( A2 = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_1190_bot_Oextremum__unique,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ bot_bo3497076220358800403od_v_v )
= ( A2 = bot_bo3497076220358800403od_v_v ) ) ).
% bot.extremum_unique
thf(fact_1191_bot_Oextremum,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% bot.extremum
thf(fact_1192_bot_Oextremum,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% bot.extremum
thf(fact_1193_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_1194_bot_Oextremum,axiom,
! [A2: $o] : ( ord_less_eq_o @ bot_bot_o @ A2 ) ).
% bot.extremum
thf(fact_1195_bot_Oextremum,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A2 ) ).
% bot.extremum
thf(fact_1196_bot_Oextremum,axiom,
! [A2: set_se8455005133513928103od_v_v] : ( ord_le4714265922333009223od_v_v @ bot_bo3497076220358800403od_v_v @ A2 ) ).
% bot.extremum
thf(fact_1197_SUP__constant,axiom,
! [A: set_v,C2: set_v] :
( ( ( A = bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [Y: v] : C2
@ A ) )
= bot_bot_set_v ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_v_set_v
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1198_SUP__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_v] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bot_set_v ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple2307003700295860064_set_v
@ ( image_2529437795422174673_set_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1199_SUP__constant,axiom,
! [C2: $o,A: set_v] :
( ( complete_Sup_Sup_o
@ ( image_v_o
@ ^ [Y: v] : C2
@ A ) )
= ( ( ( A = bot_bot_set_v )
=> bot_bot_o )
& ( ( A != bot_bot_set_v )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_1200_SUP__constant,axiom,
! [C2: $o,A: set_Product_prod_v_v] :
( ( complete_Sup_Sup_o
@ ( image_4100611349647363702_v_v_o
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= ( ( ( A = bot_bo723834152578015283od_v_v )
=> bot_bot_o )
& ( ( A != bot_bo723834152578015283od_v_v )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_1201_SUP__constant,axiom,
! [A: set_v,C2: set_Product_prod_v_v] :
( ( ( A = bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [Y: v] : C2
@ A ) )
= bot_bo723834152578015283od_v_v ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_181480991005670265od_v_v
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1202_SUP__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bo723834152578015283od_v_v ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple5788137035815166516od_v_v
@ ( image_145992280190715813od_v_v
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1203_SUP__constant,axiom,
! [A: set_v,C2: set_o] :
( ( ( A = bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [Y: v] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_v )
=> ( ( comple90263536869209701_set_o
@ ( image_v_set_o
@ ^ [Y: v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1204_SUP__constant,axiom,
! [A: set_Product_prod_v_v,C2: set_o] :
( ( ( A = bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bo723834152578015283od_v_v )
=> ( ( comple90263536869209701_set_o
@ ( image_3142478075549801814_set_o
@ ^ [Y: product_prod_v_v] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_1205_SUP__empty,axiom,
! [F: v > set_v] :
( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ bot_bot_set_v ) )
= bot_bot_set_v ) ).
% SUP_empty
thf(fact_1206_SUP__empty,axiom,
! [F: product_prod_v_v > set_v] :
( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ bot_bo723834152578015283od_v_v ) )
= bot_bot_set_v ) ).
% SUP_empty
thf(fact_1207_SUP__empty,axiom,
! [F: v > $o] :
( ( complete_Sup_Sup_o @ ( image_v_o @ F @ bot_bot_set_v ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1208_SUP__empty,axiom,
! [F: product_prod_v_v > $o] :
( ( complete_Sup_Sup_o @ ( image_4100611349647363702_v_v_o @ F @ bot_bo723834152578015283od_v_v ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1209_SUP__empty,axiom,
! [F: v > set_Product_prod_v_v] :
( ( comple5788137035815166516od_v_v @ ( image_181480991005670265od_v_v @ F @ bot_bot_set_v ) )
= bot_bo723834152578015283od_v_v ) ).
% SUP_empty
thf(fact_1210_SUP__empty,axiom,
! [F: product_prod_v_v > set_Product_prod_v_v] :
( ( comple5788137035815166516od_v_v @ ( image_145992280190715813od_v_v @ F @ bot_bo723834152578015283od_v_v ) )
= bot_bo723834152578015283od_v_v ) ).
% SUP_empty
thf(fact_1211_SUP__empty,axiom,
! [F: v > set_o] :
( ( comple90263536869209701_set_o @ ( image_v_set_o @ F @ bot_bot_set_v ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_1212_SUP__empty,axiom,
! [F: product_prod_v_v > set_o] :
( ( comple90263536869209701_set_o @ ( image_3142478075549801814_set_o @ F @ bot_bo723834152578015283od_v_v ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_1213_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_1214_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_1215_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_1216_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_1217_subset__emptyI,axiom,
! [A: set_set_o] :
( ! [X4: set_o] :
~ ( member_set_o @ X4 @ A )
=> ( ord_le4374716579403074808_set_o @ A @ bot_bot_set_set_o ) ) ).
% subset_emptyI
thf(fact_1218_subset__emptyI,axiom,
! [A: set_v] :
( ! [X4: v] :
~ ( member_v @ X4 @ A )
=> ( ord_less_eq_set_v @ A @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_1219_subset__emptyI,axiom,
! [A: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X4 @ A )
=> ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_1220_subset__emptyI,axiom,
! [A: set_o] :
( ! [X4: $o] :
~ ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1221_subset__emptyI,axiom,
! [A: set_set_v] :
( ! [X4: set_v] :
~ ( member_set_v @ X4 @ A )
=> ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v ) ) ).
% subset_emptyI
thf(fact_1222_subset__emptyI,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ! [X4: set_Product_prod_v_v] :
~ ( member8406446414694345712od_v_v @ X4 @ A )
=> ( ord_le4714265922333009223od_v_v @ A @ bot_bo3497076220358800403od_v_v ) ) ).
% subset_emptyI
thf(fact_1223_empty__Union__conv,axiom,
! [A: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_1224_empty__Union__conv,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_1225_empty__Union__conv,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_1226_Union__empty__conv,axiom,
! [A: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_1227_Union__empty__conv,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_1228_Union__empty__conv,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_1229_empty__in__Fpow,axiom,
! [A: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A ) ) ).
% empty_in_Fpow
thf(fact_1230_empty__in__Fpow,axiom,
! [A: set_v] : ( member_set_v @ bot_bot_set_v @ ( finite_Fpow_v @ A ) ) ).
% empty_in_Fpow
thf(fact_1231_empty__in__Fpow,axiom,
! [A: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ bot_bo723834152578015283od_v_v @ ( finite315275465967970893od_v_v @ A ) ) ).
% empty_in_Fpow
thf(fact_1232_hd__in__set,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( member_o @ ( hd_o @ Xs ) @ ( set_o2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1233_hd__in__set,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( Xs != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1234_hd__in__set,axiom,
! [Xs: list_set_v] :
( ( Xs != nil_set_v )
=> ( member_set_v @ ( hd_set_v @ Xs ) @ ( set_set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1235_hd__in__set,axiom,
! [Xs: list_s2556036278619108045od_v_v] :
( ( Xs != nil_se4518395726187269069od_v_v )
=> ( member8406446414694345712od_v_v @ ( hd_set3805956848269000626od_v_v @ Xs ) @ ( set_se5775211966769592924od_v_v @ Xs ) ) ) ).
% hd_in_set
thf(fact_1236_hd__in__set,axiom,
! [Xs: list_set_o] :
( ( Xs != nil_set_o )
=> ( member_set_o @ ( hd_set_o @ Xs ) @ ( set_set_o2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1237_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1238_list_Oset__sel_I1_J,axiom,
! [A2: list_o] :
( ( A2 != nil_o )
=> ( member_o @ ( hd_o @ A2 ) @ ( set_o2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1239_list_Oset__sel_I1_J,axiom,
! [A2: list_P7986770385144383213od_v_v] :
( ( A2 != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A2 ) @ ( set_Product_prod_v_v2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1240_list_Oset__sel_I1_J,axiom,
! [A2: list_set_v] :
( ( A2 != nil_set_v )
=> ( member_set_v @ ( hd_set_v @ A2 ) @ ( set_set_v2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1241_list_Oset__sel_I1_J,axiom,
! [A2: list_s2556036278619108045od_v_v] :
( ( A2 != nil_se4518395726187269069od_v_v )
=> ( member8406446414694345712od_v_v @ ( hd_set3805956848269000626od_v_v @ A2 ) @ ( set_se5775211966769592924od_v_v @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1242_list_Oset__sel_I1_J,axiom,
! [A2: list_set_o] :
( ( A2 != nil_set_o )
=> ( member_set_o @ ( hd_set_o @ A2 ) @ ( set_set_o2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1243_list_Oset__sel_I1_J,axiom,
! [A2: list_v] :
( ( A2 != nil_v )
=> ( member_v @ ( hd_v @ A2 ) @ ( set_v2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1244_finite__has__maximal,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A )
=> ( ( ord_less_eq_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1245_finite__has__maximal,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1246_finite__has__maximal,axiom,
! [A: set_set_o] :
( ( finite_finite_set_o @ A )
=> ( ( A != bot_bot_set_set_o )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1247_finite__has__maximal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1248_finite__has__maximal,axiom,
! [A: set_set_set_v] :
( ( finite8701002811114149628_set_v @ A )
=> ( ( A != bot_bo5775917114081396255_set_v )
=> ? [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
& ! [Xa: set_set_v] :
( ( member_set_set_v @ Xa @ A )
=> ( ( ord_le5216385588623774835_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1249_finite__has__maximal,axiom,
! [A: set_se2157405561750842759od_v_v] :
( ( finite2944697185239447248od_v_v @ A )
=> ( ( A != bot_bo2111273119081350131od_v_v )
=> ? [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
& ! [Xa: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Xa @ A )
=> ( ( ord_le4714265922333009223od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1250_finite__has__minimal,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A )
=> ( ( ord_less_eq_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1251_finite__has__minimal,axiom,
! [A: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1252_finite__has__minimal,axiom,
! [A: set_set_o] :
( ( finite_finite_set_o @ A )
=> ( ( A != bot_bot_set_set_o )
=> ? [X4: set_o] :
( ( member_set_o @ X4 @ A )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1253_finite__has__minimal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1254_finite__has__minimal,axiom,
! [A: set_set_set_v] :
( ( finite8701002811114149628_set_v @ A )
=> ( ( A != bot_bo5775917114081396255_set_v )
=> ? [X4: set_set_v] :
( ( member_set_set_v @ X4 @ A )
& ! [Xa: set_set_v] :
( ( member_set_set_v @ Xa @ A )
=> ( ( ord_le5216385588623774835_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1255_finite__has__minimal,axiom,
! [A: set_se2157405561750842759od_v_v] :
( ( finite2944697185239447248od_v_v @ A )
=> ( ( A != bot_bo2111273119081350131od_v_v )
=> ? [X4: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ X4 @ A )
& ! [Xa: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ Xa @ A )
=> ( ( ord_le4714265922333009223od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1256_less__eq__Sup,axiom,
! [A: set_set_set_v,U: set_set_v] :
( ! [V3: set_set_v] :
( ( member_set_set_v @ V3 @ A )
=> ( ord_le5216385588623774835_set_v @ U @ V3 ) )
=> ( ( A != bot_bo5775917114081396255_set_v )
=> ( ord_le5216385588623774835_set_v @ U @ ( comple5450237519782573632_set_v @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1257_less__eq__Sup,axiom,
! [A: set_se2157405561750842759od_v_v,U: set_se8455005133513928103od_v_v] :
( ! [V3: set_se8455005133513928103od_v_v] :
( ( member5511408251247217616od_v_v @ V3 @ A )
=> ( ord_le4714265922333009223od_v_v @ U @ V3 ) )
=> ( ( A != bot_bo2111273119081350131od_v_v )
=> ( ord_le4714265922333009223od_v_v @ U @ ( comple6800410395631251988od_v_v @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1258_less__eq__Sup,axiom,
! [A: set_set_v,U: set_v] :
( ! [V3: set_v] :
( ( member_set_v @ V3 @ A )
=> ( ord_less_eq_set_v @ U @ V3 ) )
=> ( ( A != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ U @ ( comple2307003700295860064_set_v @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1259_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V3: $o] :
( ( member_o @ V3 @ A )
=> ( ord_less_eq_o @ U @ V3 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1260_less__eq__Sup,axiom,
! [A: set_se8455005133513928103od_v_v,U: set_Product_prod_v_v] :
( ! [V3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ V3 @ A )
=> ( ord_le7336532860387713383od_v_v @ U @ V3 ) )
=> ( ( A != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ U @ ( comple5788137035815166516od_v_v @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1261_less__eq__Sup,axiom,
! [A: set_set_o,U: set_o] :
( ! [V3: set_o] :
( ( member_set_o @ V3 @ A )
=> ( ord_less_eq_set_o @ U @ V3 ) )
=> ( ( A != bot_bot_set_set_o )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1262_cSup__eq__non__empty,axiom,
! [X3: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( X3 != bot_bo3497076220358800403od_v_v )
=> ( ! [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ A2 ) )
=> ( ! [Y3: set_Product_prod_v_v] :
( ! [X: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X @ X3 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y3 ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ Y3 ) )
=> ( ( comple5788137035815166516od_v_v @ X3 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1263_cSup__eq__non__empty,axiom,
! [X3: set_set_o,A2: set_o] :
( ( X3 != bot_bot_set_set_o )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ X3 )
=> ( ord_less_eq_set_o @ X4 @ A2 ) )
=> ( ! [Y3: set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ X3 )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ord_less_eq_set_o @ A2 @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ X3 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1264_cc__def,axiom,
( cc
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) ) ).
% cc_def
thf(fact_1265_pfx_I3_J,axiom,
( ( sCC_Bl1280885523602775798t_unit @ e2 )
= ( ^ [X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) ) ) ) ).
% pfx(3)
thf(fact_1266__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pfx_O_A_092_060lbrakk_062stack_Ae_A_061_Apfx_A_064_Astack_Ae_H_059_Astack_Ae_H_A_092_060noteq_062_A_091_093_059_Alet_Acc_A_061_A_092_060Union_062_A_123_092_060S_062_Ae_An_A_124n_O_An_A_092_060in_062_Aset_Apfx_A_092_060union_062_A_123hd_A_Istack_Ae_H_J_125_125_Ain_A_092_060S_062_Ae_H_A_061_A_I_092_060lambda_062x_O_Aif_Ax_A_092_060in_062_Acc_Athen_Acc_Aelse_A_092_060S_062_Ae_Ax_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pfx: list_v] :
( ( ( sCC_Bl8828226123343373779t_unit @ e )
= ( append_v @ Pfx @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v )
=> ( ( sCC_Bl1280885523602775798t_unit @ e2 )
!= ( ^ [X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ N2 ) )
& ( member_v @ N2 @ ( sup_sup_set_v @ ( set_v2 @ Pfx ) @ ( insert_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) @ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>pfx. \<lbrakk>stack e = pfx @ stack e'; stack e' \<noteq> []; let cc = \<Union> {\<S> e n |n. n \<in> set pfx \<union> {hd (stack e')}} in \<S> e' = (\<lambda>x. if x \<in> cc then cc else \<S> e x)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1267_subscc__add,axiom,
! [S: set_v,X5: v,Y2: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X5 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X5 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y2 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_1268_ra__add__edge,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1269_ra__cases,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( ( X5 = Y2 )
| ? [Z6: v] :
( ( member_v @ Z6 @ ( successors @ X5 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X5 @ Z6 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z6 @ Y2 @ E4 ) ) ) ) ).
% ra_cases
thf(fact_1270_edge__ra,axiom,
! [Y2: v,X5: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y2 @ ( successors @ X5 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X5 @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 ) ) ) ).
% edge_ra
thf(fact_1271_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X2: v,E8: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A32 = E8 ) )
| ? [X2: v,Y: v,E8: set_Product_prod_v_v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( A32 = E8 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y @ E8 )
& ( member_v @ Z5 @ ( successors @ Y ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z5 ) @ E8 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1272_ra__succ,axiom,
! [X5: v,Y2: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Y2 @ E4 )
=> ( ( member_v @ Z2 @ ( successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X5 @ Z2 @ E4 ) ) ) ) ).
% ra_succ
thf(fact_1273_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y3 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1274_scc__partition,axiom,
! [S: set_v,S4: set_v,X5: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
=> ( ( member_v @ X5 @ ( inf_inf_set_v @ S @ S4 ) )
=> ( S = S4 ) ) ) ) ).
% scc_partition
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X5: set_v,Y2: set_v] :
( ( if_set_v @ $false @ X5 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X5: set_v,Y2: set_v] :
( ( if_set_v @ $true @ X5 @ Y2 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_set_v @ cc
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [X2: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ e @ X2 ) )
& ( member_v @ X2 @ cc ) ) ) ) ) ).
%------------------------------------------------------------------------------