TPTP Problem File: SLH0956^1.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 : Rewrite_Properties_Reduction/0020_Rewriting_Properties/prob_00545_025035__14193874_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1606 ( 444 unt; 330 typ; 0 def)
% Number of atoms : 3844 ( 945 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 14062 ( 415 ~; 35 |; 223 &;11483 @)
% ( 0 <=>;1906 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 720 ( 720 >; 0 *; 0 +; 0 <<)
% Number of symbols : 297 ( 296 usr; 29 con; 0-5 aty)
% Number of variables : 3654 ( 112 ^;3427 !; 115 ?;3654 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:20.515
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
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thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
lex_pr5733628728943162184rm_a_b: set_Pr1261947904930325089at_nat > set_Pr2972776593051762503rm_a_b > set_Pr4403016048539538589rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr2499763052546576405rm_a_b: set_Pr1261947904930325089at_nat > set_Pr4386577575007340137rm_a_b > set_Pr5038301440468608839rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Nat__Onat,type,
lex_pr3675074321942177898_b_nat: set_Pr2972776593051762503rm_a_b > set_Pr1261947904930325089at_nat > set_Pr3273496522534110801_b_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
lex_pr3939907742592656111rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b > set_Pr6848273517546626247rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr7567299705543466414rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr6651502721622080345rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
lex_pr3461217179711548565_b_nat: set_Pr4386577575007340137rm_a_b > set_Pr1261947904930325089at_nat > set_Pr4200871855241742151_b_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
lex_pr5735081776509173764rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b > set_Pr8337550613148266901rm_a_b ).
thf(sy_c_Wellfounded_Olex__prod_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_pr6163557265797435481rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Wellfounded_Omax__ext_001t__Nat__Onat,type,
max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Omin__ext_001t__Nat__Onat,type,
min_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
wf_Pro2335863617654816626rm_a_b: set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_Wellfounded_Owf_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
wf_term_a_b: set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Wfrec_Osame__fst_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
same_f843731131676418198rm_a_b: ( term_a_b > $o ) > ( term_a_b > set_Pr4386577575007340137rm_a_b ) > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8823642877475723282rm_a_b: set_Pr4386577575007340137rm_a_b > set_se1467346398288745161rm_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member_set_term_a_b: set_term_a_b > set_set_term_a_b > $o ).
thf(sy_c_member_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: term_a_b > set_term_a_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v__092_060S_062,type,
s: set_Pr4386577575007340137rm_a_b ).
% Relevant facts (1275)
thf(fact_0_assms_I3_J,axiom,
( ( abstract_NF_term_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) )
= ( abstract_NF_term_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ s ) ) ) ) ).
% assms(3)
thf(fact_1_NE__symmetric,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( rewriting_NE_on_nat @ R @ S @ top_top_set_nat )
=> ( rewriting_NE_on_nat @ S @ R @ top_top_set_nat ) ) ).
% NE_symmetric
thf(fact_2_NE__symmetric,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( rewrit4650708409860811906rm_a_b @ R @ S @ top_to1314267278846557113rm_a_b )
=> ( rewrit4650708409860811906rm_a_b @ S @ R @ top_to1314267278846557113rm_a_b ) ) ).
% NE_symmetric
thf(fact_3_NE__symmetric,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( rewrit7370782028743737947rm_a_b @ R @ S @ top_top_set_term_a_b )
=> ( rewrit7370782028743737947rm_a_b @ S @ R @ top_top_set_term_a_b ) ) ).
% NE_symmetric
thf(fact_4_UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_6_UNIV__I,axiom,
! [X: term_a_b] : ( member_term_a_b @ X @ top_top_set_term_a_b ) ).
% UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X @ top_to1314267278846557113rm_a_b ) ).
% iso_tuple_UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
! [X: term_a_b] : ( member_term_a_b @ X @ top_top_set_term_a_b ) ).
% iso_tuple_UNIV_I
thf(fact_10_assms_I2_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ f @ s ) )
=> ( rewrit3451484132325884455dp_a_b @ f @ s @ r @ S @ T ) ) ).
% assms(2)
thf(fact_11_assms_I1_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ f @ r ) )
=> ( rewrit3451484132325884455dp_a_b @ f @ r @ s @ S @ T ) ) ).
% assms(1)
thf(fact_12_CE__symmetric,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( rewriting_CE_on_nat @ R @ S @ top_top_set_nat )
=> ( rewriting_CE_on_nat @ S @ R @ top_top_set_nat ) ) ).
% CE_symmetric
thf(fact_13_CE__symmetric,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( rewrit1455736940331458423rm_a_b @ R @ S @ top_to1314267278846557113rm_a_b )
=> ( rewrit1455736940331458423rm_a_b @ S @ R @ top_to1314267278846557113rm_a_b ) ) ).
% CE_symmetric
thf(fact_14_CE__symmetric,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( rewrit660295756789745446rm_a_b @ R @ S @ top_top_set_term_a_b )
=> ( rewrit660295756789745446rm_a_b @ S @ R @ top_top_set_term_a_b ) ) ).
% CE_symmetric
thf(fact_15_UN__redp__def,axiom,
( rewrit983855061078472681dp_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: term_a_b,T2: term_a_b] :
( ( ( member_term_a_b @ S2 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
& ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) )
=> ( S2 = T2 ) ) ) ) ).
% UN_redp_def
thf(fact_16_UNIV__eq__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( top_to1314267278846557113rm_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_17_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_18_UNIV__eq__I,axiom,
! [A: set_term_a_b] :
( ! [X2: term_a_b] : ( member_term_a_b @ X2 @ A )
=> ( top_top_set_term_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_19_UNIV__witness,axiom,
? [X2: term_a_b] : ( member_term_a_b @ X2 @ top_top_set_term_a_b ) ).
% UNIV_witness
thf(fact_20_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_21_UNIV__witness,axiom,
? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ top_to1314267278846557113rm_a_b ) ).
% UNIV_witness
thf(fact_22_Sup__UNIV,axiom,
( ( comple5064467279716934173rm_a_b @ top_to5470816188456996400rm_a_b )
= top_top_set_term_a_b ) ).
% Sup_UNIV
thf(fact_23_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_24_Sup__UNIV,axiom,
( ( comple4970752389892389974rm_a_b @ top_to8981750078293510169rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% Sup_UNIV
thf(fact_25_all__ctxt__closed__sig__rsteps,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ).
% all_ctxt_closed_sig_rsteps
thf(fact_26_atMost__eq__UNIV__iff,axiom,
! [X: set_term_a_b] :
( ( ( set_or8375481205460978499rm_a_b @ X )
= top_to5470816188456996400rm_a_b )
= ( X = top_top_set_term_a_b ) ) ).
% atMost_eq_UNIV_iff
thf(fact_27_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_28_atMost__eq__UNIV__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( ( set_or8794634880352032432rm_a_b @ X )
= top_to8981750078293510169rm_a_b )
= ( X = top_to1314267278846557113rm_a_b ) ) ).
% atMost_eq_UNIV_iff
thf(fact_29_atMost__UNIV__triv,axiom,
( ( set_or8375481205460978499rm_a_b @ top_top_set_term_a_b )
= top_to5470816188456996400rm_a_b ) ).
% atMost_UNIV_triv
thf(fact_30_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_31_atMost__UNIV__triv,axiom,
( ( set_or8794634880352032432rm_a_b @ top_to1314267278846557113rm_a_b )
= top_to8981750078293510169rm_a_b ) ).
% atMost_UNIV_triv
thf(fact_32_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_33_top__set__def,axiom,
( top_top_set_term_a_b
= ( collect_term_a_b @ top_top_term_a_b_o ) ) ).
% top_set_def
thf(fact_34_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_35_top__set__def,axiom,
( top_to1314267278846557113rm_a_b
= ( collec99177395925924084rm_a_b @ top_to1262317115727556708_a_b_o ) ) ).
% top_set_def
thf(fact_36_Union__UNIV,axiom,
( ( comple5064467279716934173rm_a_b @ top_to5470816188456996400rm_a_b )
= top_top_set_term_a_b ) ).
% Union_UNIV
thf(fact_37_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_38_Union__UNIV,axiom,
( ( comple4970752389892389974rm_a_b @ top_to8981750078293510169rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% Union_UNIV
thf(fact_39_not__UNIV__eq__Iic,axiom,
! [H: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_40_srsteps__with__root__step__sresteps__eqD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_with_root_step_sresteps_eqD
thf(fact_41_NE__redp__def,axiom,
( rewrit3451484132325884455dp_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,S2: term_a_b,T2: term_a_b] :
( ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) ) ) ) ) ) ).
% NE_redp_def
thf(fact_42_NF__I,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ! [B: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B ) @ A )
=> ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ).
% NF_I
thf(fact_43_NF__I,axiom,
! [A2: nat,A: set_Pr1261947904930325089at_nat] :
( ! [B: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ A )
=> ( member_nat @ A2 @ ( abstract_NF_nat @ A ) ) ) ).
% NF_I
thf(fact_44_NF__I,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ A )
=> ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_I
thf(fact_45_cSup__atMost,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_46_rstep__ruleI,axiom,
! [L: term_a_b,R: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep_ruleI
thf(fact_47_NFP__redp__def,axiom,
( rewrit232733863534887208dp_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S2: term_a_b,T2: term_a_b] :
( ( member_term_a_b @ T2 @ ( abstract_NF_term_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T2 ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% NFP_redp_def
thf(fact_48_NFP__stepD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ C @ ( abstract_NF_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_49_NFP__stepD,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat,C: nat] :
( ( rewriting_NFP_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member_nat @ C @ ( abstract_NF_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_50_NFP__stepD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( rewrit2138790387785322561rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ C @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).
% NFP_stepD
thf(fact_51_NF__not__suc,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_52_NF__not__suc,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member_nat @ X @ ( abstract_NF_nat @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_53_NF__not__suc,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ).
% NF_not_suc
thf(fact_54_r__into__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% r_into_rtrancl
thf(fact_55_prod_Oinject,axiom,
! [X1: term_a_b,X22: term_a_b,Y1: term_a_b,Y2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X22 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_57_NF__E,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_58_NF__E,axiom,
! [A2: nat,A: set_Pr1261947904930325089at_nat,B2: nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ A ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_59_NF__E,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A ) ) ).
% NF_E
thf(fact_60_UnionI,axiom,
! [X3: set_Pr4386577575007340137rm_a_b,C2: set_se1467346398288745161rm_a_b,A: produc357393685978478089rm_a_b] :
( ( member8823642877475723282rm_a_b @ X3 @ C2 )
=> ( ( member5869715511025134514rm_a_b @ A @ X3 )
=> ( member5869715511025134514rm_a_b @ A @ ( comple4970752389892389974rm_a_b @ C2 ) ) ) ) ).
% UnionI
thf(fact_61_UnionI,axiom,
! [X3: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X3 @ C2 )
=> ( ( member_nat @ A @ X3 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_62_UnionI,axiom,
! [X3: set_term_a_b,C2: set_set_term_a_b,A: term_a_b] :
( ( member_set_term_a_b @ X3 @ C2 )
=> ( ( member_term_a_b @ A @ X3 )
=> ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) ) ) ) ).
% UnionI
thf(fact_63_Union__iff,axiom,
! [A: produc357393685978478089rm_a_b,C2: set_se1467346398288745161rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( comple4970752389892389974rm_a_b @ C2 ) )
= ( ? [X4: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ X4 @ C2 )
& ( member5869715511025134514rm_a_b @ A @ X4 ) ) ) ) ).
% Union_iff
thf(fact_64_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ C2 )
& ( member_nat @ A @ X4 ) ) ) ) ).
% Union_iff
thf(fact_65_Union__iff,axiom,
! [A: term_a_b,C2: set_set_term_a_b] :
( ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) )
= ( ? [X4: set_term_a_b] :
( ( member_set_term_a_b @ X4 @ C2 )
& ( member_term_a_b @ A @ X4 ) ) ) ) ).
% Union_iff
thf(fact_66_UnionE,axiom,
! [A: produc357393685978478089rm_a_b,C2: set_se1467346398288745161rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ ( comple4970752389892389974rm_a_b @ C2 ) )
=> ~ ! [X5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A @ X5 )
=> ~ ( member8823642877475723282rm_a_b @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_67_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X5: set_nat] :
( ( member_nat @ A @ X5 )
=> ~ ( member_set_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_68_UnionE,axiom,
! [A: term_a_b,C2: set_set_term_a_b] :
( ( member_term_a_b @ A @ ( comple5064467279716934173rm_a_b @ C2 ) )
=> ~ ! [X5: set_term_a_b] :
( ( member_term_a_b @ A @ X5 )
=> ~ ( member_set_term_a_b @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_69_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: term_a_b,B: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A4 @ B ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_70_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A4: term_a_b,B: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A4 @ B ) ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_71_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: term_a_b,B: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A4 @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_72_mem__Collect__eq,axiom,
! [A2: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A2 @ ( collec99177395925924084rm_a_b @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A2: term_a_b,P2: term_a_b > $o] :
( ( member_term_a_b @ A2 @ ( collect_term_a_b @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_term_a_b] :
( ( collect_term_a_b
@ ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Pair__inject,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A2 @ B2 )
= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_79_prod__cases,axiom,
! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
( ! [A4: term_a_b,B: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A4 @ B ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_80_surj__pair,axiom,
! [P: produc357393685978478089rm_a_b] :
? [X2: term_a_b,Y3: term_a_b] :
( P
= ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_81_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A4: term_a_b,B: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A4 @ B ) ) ).
% old.prod.exhaust
thf(fact_82_converse__rtrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_83_converse__rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P2 @ B2 )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_84_converse__rtranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_85_rtrancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( P2 @ A2 )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_86_rtrancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_87_rtranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A2 != B2 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_88_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_89_rtrancl_Ortrancl__refl,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_90_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
= ( ? [A5: term_a_b] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_91_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_92_NF__iff__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
= ( ! [B4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_93_NF__iff__no__step,axiom,
! [A2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ A ) )
= ( ! [B4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_94_NF__iff__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
= ( ! [B4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B4 ) @ A ) ) ) ).
% NF_iff_no_step
thf(fact_95_NF__no__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B5: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ A ) ) ).
% NF_no_step
thf(fact_96_NF__no__step,axiom,
! [A2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ A ) )
=> ! [B5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ A ) ) ).
% NF_no_step
thf(fact_97_NF__no__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ A ) ) ).
% NF_no_step
thf(fact_98_top__empty__eq,axiom,
( top_top_term_a_b_o
= ( ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ top_top_set_term_a_b ) ) ) ).
% top_empty_eq
thf(fact_99_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_100_top__empty__eq,axiom,
( top_to1262317115727556708_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ top_to1314267278846557113rm_a_b ) ) ) ).
% top_empty_eq
thf(fact_101_meetI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) ) ) ) ).
% meetI
thf(fact_102_normalizability__I,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_103_normalizability__I,axiom,
! [A2: nat,B2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ A ) )
=> ( ( member_nat @ B2 @ ( abstract_NF_nat @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra7312431243108365096ty_nat @ A ) ) ) ) ).
% normalizability_I
thf(fact_104_normalizability__I,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I
thf(fact_105_meetE,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ~ ! [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetE
thf(fact_106_meetD,axiom,
! [B2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra1093061187967292567rm_a_b @ A ) )
=> ? [A4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% meetD
thf(fact_107_NFP__on__def,axiom,
( rewrit6304709981509430748rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ) ) ).
% NFP_on_def
thf(fact_108_srrstep__to__srestep,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ).
% srrstep_to_srestep
thf(fact_109_rrstep__basicI,axiom,
! [L: term_a_b,R: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ ( rrstep_a_b @ R3 ) ) ) ).
% rrstep_basicI
thf(fact_110_normalizability__I_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% normalizability_I'
thf(fact_111_NE__on__def,axiom,
( rewrit7370782028743737947rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ S2 ) ) ) ) ) ) ).
% NE_on_def
thf(fact_112_normalizability__E,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ A ) )
=> ~ ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_113_normalizability__E,axiom,
! [A2: nat,B2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra7312431243108365096ty_nat @ A ) )
=> ~ ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ A ) )
=> ~ ( member_nat @ B2 @ ( abstract_NF_nat @ A ) ) ) ) ).
% normalizability_E
thf(fact_114_normalizability__E,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ A ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) ) ) ) ).
% normalizability_E
thf(fact_115_UNF__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C4 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B = C4 ) ) ) )
=> ( abstra1303798319023342519rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_116_UNF__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat,B: nat,C4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ ( abstra7312431243108365096ty_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ C4 ) @ ( abstra7312431243108365096ty_nat @ R ) )
=> ( B = C4 ) ) ) )
=> ( abstract_UNF_on_nat @ R @ A ) ) ).
% UNF_onI
thf(fact_117_UNF__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A4 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C4 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B = C4 ) ) ) )
=> ( abstra3619855735679963878rm_a_b @ R @ A ) ) ).
% UNF_onI
thf(fact_118_WN__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A )
=> ? [B5: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B5 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
=> ( abstra2721873542170697047rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_119_WN__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B5 ) @ ( abstra7312431243108365096ty_nat @ R ) ) )
=> ( abstract_WN_on_nat @ R @ A ) ) ).
% WN_onI
thf(fact_120_WN__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
( ( member_term_a_b @ A4 @ A )
=> ? [B5: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B5 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( abstra4644735066934440262rm_a_b @ R @ A ) ) ).
% WN_onI
thf(fact_121_srsteps__with__root__step__def,axiom,
( srstep7844470518422762656ep_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b] : ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) @ ( relcom370159955682700863rm_a_b @ ( sig_step_a_b @ F @ ( rrstep_a_b @ R2 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ).
% srsteps_with_root_step_def
thf(fact_122_UNF__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_123_UNF__onD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_UNF_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra7312431243108365096ty_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( abstra7312431243108365096ty_nat @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_124_UNF__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ( B2 = C ) ) ) ) ) ).
% UNF_onD
thf(fact_125_UNF__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_126_UNF__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_UNF_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra7312431243108365096ty_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_127_UNF__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( B2 != C )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ) ).
% UNF_onE
thf(fact_128_UNF__on__def,axiom,
( abstra3619855735679963878rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) )
=> ( B4 = C3 ) ) ) ) ) ).
% UNF_on_def
thf(fact_129_same__fstI,axiom,
! [P2: term_a_b > $o,X: term_a_b,Y4: term_a_b,Y: term_a_b,R5: term_a_b > set_Pr4386577575007340137rm_a_b] :
( ( P2 @ X )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y4 @ Y ) @ ( R5 @ X ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y4 ) @ ( produc7020197800436672577rm_a_b @ X @ Y ) ) @ ( same_f843731131676418198rm_a_b @ P2 @ R5 ) ) ) ) ).
% same_fstI
thf(fact_130_some__NF__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( Y
= ( abstra2589370493620291516rm_a_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_131_some__NF__UNF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( abstract_UNF_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member_nat @ Y @ ( abstract_NF_nat @ R ) )
=> ( Y
= ( abstract_some_NF_nat @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_132_some__NF__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( Y
= ( abstra8203652949708803681rm_a_b @ R @ X ) ) ) ) ) ).
% some_NF_UNF
thf(fact_133_WN__on__def,axiom,
( abstra4644735066934440262rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ? [B4: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra2972651231392952684rm_a_b @ R4 ) ) ) ) ) ).
% WN_on_def
thf(fact_134_WN__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ~ ! [B: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_135_WN__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
( ( abstract_WN_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ~ ! [B: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ) ).
% WN_onE
thf(fact_136_WN__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ~ ! [B: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onE
thf(fact_137_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A: set_Pr4386577575007340137rm_a_b,S: term_a_b,B6: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B6 @ ( relcom370159955682700863rm_a_b @ A @ B6 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_138_relcompEpair,axiom,
! [A2: term_a_b,C: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_139_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [X2: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X2 @ Z ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ S ) ) ) ) ).
% relcompE
thf(fact_140_relcomp_OrelcompI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( relcom370159955682700863rm_a_b @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_141_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
= ( ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_142_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R @ S ) )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_143_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T3: term_a_b,U2: term_a_b,R6: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T3 )
=> ( ( Q @ R6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ R6 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T3 ) @ ( F3 @ U2 ) ) @ ( G @ R6 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R5 )
=> ( ( Q @ S4 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S4 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R5 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_144_WN__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_145_WN__onD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
( ( abstract_WN_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ) ).
% WN_onD
thf(fact_146_WN__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ? [B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ).
% WN_onD
thf(fact_147_UNC__imp__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4649771476585718372rm_a_b @ R )
=> ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% UNC_imp_UNF
thf(fact_148_UNC__imp__UNF,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_UNC_nat @ R )
=> ( abstract_UNF_on_nat @ R @ top_top_set_nat ) ) ).
% UNC_imp_UNF
thf(fact_149_UNC__imp__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ R )
=> ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% UNC_imp_UNF
thf(fact_150_WN__UNF__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4644735066934440262rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ).
% WN_UNF_imp_CR
thf(fact_151_WN__UNF__imp__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_WN_on_nat @ R @ top_top_set_nat )
=> ( ( abstract_UNF_on_nat @ R @ top_top_set_nat )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ) ).
% WN_UNF_imp_CR
thf(fact_152_WN__UNF__imp__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra2721873542170697047rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ) ).
% WN_UNF_imp_CR
thf(fact_153_UNF__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R3 @ S @ T ) ) ) ).
% UNF_to_rrstep
thf(fact_154_srsteps__with__root__step__srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_with_root_step_srstepsD
thf(fact_155_all__ctxt__closed__srstep__conversions,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] : ( terms_5226143800768910156ed_a_b @ F2 @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ).
% all_ctxt_closed_srstep_conversions
thf(fact_156_NF__join__imp__reach,axiom,
! [Y: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ A ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_157_NF__join__imp__reach,axiom,
! [Y: nat,A: set_Pr1261947904930325089at_nat,X: nat] :
( ( member_nat @ Y @ ( abstract_NF_nat @ A ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( abstract_join_nat @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_158_NF__join__imp__reach,axiom,
! [Y: term_a_b,A: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% NF_join_imp_reach
thf(fact_159_commute__redp__def,axiom,
( rewrit8865997592796547114dp_a_b
= ( ^ [F: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,S2: term_a_b,T2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S2 @ T2 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ S3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F @ ( rstep_a_b @ R2 ) ) ) ) ) ) ) ) ).
% commute_redp_def
thf(fact_160_asymI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R ) )
=> ( asym_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% asymI
thf(fact_161_asymI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R ) )
=> ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% asymI
thf(fact_162_asymI,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R ) )
=> ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% asymI
thf(fact_163_NFP__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F2 @ R3 @ S @ T ) ) ) ).
% NFP_to_rrstep
thf(fact_164_Un__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B6 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
| ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_165_Un__iff,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B6 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_166_Un__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B6 ) )
= ( ( member_term_a_b @ C @ A )
| ( member_term_a_b @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_167_UnCI,axiom,
! [C: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C @ B6 )
=> ( member5869715511025134514rm_a_b @ C @ A ) )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B6 ) ) ) ).
% UnCI
thf(fact_168_UnCI,axiom,
! [C: nat,B6: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B6 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B6 ) ) ) ).
% UnCI
thf(fact_169_UnCI,axiom,
! [C: term_a_b,B6: set_term_a_b,A: set_term_a_b] :
( ( ~ ( member_term_a_b @ C @ B6 )
=> ( member_term_a_b @ C @ A ) )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B6 ) ) ) ).
% UnCI
thf(fact_170_converse__UNIV,axiom,
( ( conver7608559976937845093rm_a_b @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% converse_UNIV
thf(fact_171_converse__iff,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converse_iff
thf(fact_172_asym__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R ) ) ) )
=> ( asym_o1443355727950249531rm_a_b @ A @ R ) ) ).
% asym_onI
thf(fact_173_asym__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R ) ) ) )
=> ( asym_on_nat @ A @ R ) ) ).
% asym_onI
thf(fact_174_asym__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ( member_term_a_b @ Y3 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R ) ) ) )
=> ( asym_on_term_a_b @ A @ R ) ) ).
% asym_onI
thf(fact_175_conversionI_H,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI'
thf(fact_176_joinI,axiom,
! [A2: term_a_b,C: term_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% joinI
thf(fact_177_joinI__left,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_left
thf(fact_178_joinI__right,axiom,
! [B2: term_a_b,A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% joinI_right
thf(fact_179_CR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C4 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra7359694574113289279rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_180_CR__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat,B: nat,C4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ C4 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C4 ) @ ( abstract_join_nat @ R ) ) ) ) )
=> ( abstract_CR_on_nat @ R @ A ) ) ).
% CR_onI
thf(fact_181_CR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A4 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ).
% CR_onI
thf(fact_182_conversionI,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ).
% conversionI
thf(fact_183_sig__step__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ R3 ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ R3 ) ) )
= ( sig_step_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% sig_step_symcl_dist
thf(fact_184_CR__imp__conversionIff__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra1130568886744591910rm_a_b @ R )
= ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_185_CR__imp__conversionIff__join,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( ( abstra2375765306220057826on_nat @ R )
= ( abstract_join_nat @ R ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_186_CR__imp__conversionIff__join,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( abstra6125656530262095991rm_a_b @ R )
= ( abstra4889682940153293507rm_a_b @ R ) ) ) ).
% CR_imp_conversionIff_join
thf(fact_187_r__into__trancl_H,axiom,
! [P: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ R )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% r_into_trancl'
thf(fact_188_UnI2,axiom,
! [C: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ B6 )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B6 ) ) ) ).
% UnI2
thf(fact_189_UnI2,axiom,
! [C: nat,B6: set_nat,A: set_nat] :
( ( member_nat @ C @ B6 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B6 ) ) ) ).
% UnI2
thf(fact_190_UnI2,axiom,
! [C: term_a_b,B6: set_term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ C @ B6 )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B6 ) ) ) ).
% UnI2
thf(fact_191_UnI1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B6 ) ) ) ).
% UnI1
thf(fact_192_UnI1,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B6 ) ) ) ).
% UnI1
thf(fact_193_UnI1,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B6 ) ) ) ).
% UnI1
thf(fact_194_UnE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( sup_su6776935440552674877rm_a_b @ A @ B6 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% UnE
thf(fact_195_UnE,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B6 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B6 ) ) ) ).
% UnE
thf(fact_196_UnE,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( sup_sup_set_term_a_b @ A @ B6 ) )
=> ( ~ ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B6 ) ) ) ).
% UnE
thf(fact_197_rstep__smycl__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( rstep_a_b @ R3 ) @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) ) )
= ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% rstep_smycl_dist
thf(fact_198_trancl__converseI,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% trancl_converseI
thf(fact_199_trancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( conver7608559976937845093rm_a_b @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_converseD
thf(fact_200_conversionE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A @ ( conver7608559976937845093rm_a_b @ A ) ) ) ) ) ).
% conversionE
thf(fact_201_CR__imp__UNC,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra4649771476585718372rm_a_b @ R ) ) ).
% CR_imp_UNC
thf(fact_202_CR__imp__UNC,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( abstract_UNC_nat @ R ) ) ).
% CR_imp_UNC
thf(fact_203_CR__imp__UNC,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( abstra8480497335821202873rm_a_b @ R ) ) ).
% CR_imp_UNC
thf(fact_204_symcl__srstep__conversion,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% symcl_srstep_conversion
thf(fact_205_srstep__symcl__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ).
% srstep_symcl_dist
thf(fact_206_trancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ R )
=> ~ ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_207_trancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7922773638565587891rm_a_b @ R ) )
= ( ? [A5: term_a_b,B4: term_a_b] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R ) )
| ? [A5: term_a_b,B4: term_a_b,C3: term_a_b] :
( ( A1 = A5 )
& ( A22 = C3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_208_trancl_Or__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_209_tranclE,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C4 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_210_trancl__trans,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_trans
thf(fact_211_trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_212_r__r__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R5: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R5 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_213_converse__tranclE,axiom,
! [X: term_a_b,Z2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_214_irrefl__trancl__rD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X2: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_215_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_216_trancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_217_trancl__trans__induct,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ( P2 @ X2 @ Y3 ) )
=> ( ! [X2: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P2 @ X2 @ Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X2 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_218_converse__trancl__induct,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B2 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_219_converse_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_220_converse_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( conver7608559976937845093rm_a_b @ R ) )
= ( ? [A5: term_a_b,B4: term_a_b] :
( ( A1 = B4 )
& ( A22 = A5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_221_converseD,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( conver7608559976937845093rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ).
% converseD
thf(fact_222_converseE,axiom,
! [Yx: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Yx @ ( conver7608559976937845093rm_a_b @ R ) )
=> ~ ! [X2: term_a_b,Y3: term_a_b] :
( ( Yx
= ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R ) ) ) ).
% converseE
thf(fact_223_converseI,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% converseI
thf(fact_224_sig__step__conversion__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) ) ).
% sig_step_conversion_dist
thf(fact_225_Un__UNIV__right,axiom,
! [A: set_term_a_b] :
( ( sup_sup_set_term_a_b @ A @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% Un_UNIV_right
thf(fact_226_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_227_Un__UNIV__right,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_right
thf(fact_228_Un__UNIV__left,axiom,
! [B6: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ B6 )
= top_top_set_term_a_b ) ).
% Un_UNIV_left
thf(fact_229_Un__UNIV__left,axiom,
! [B6: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B6 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_230_Un__UNIV__left,axiom,
! [B6: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ B6 )
= top_to1314267278846557113rm_a_b ) ).
% Un_UNIV_left
thf(fact_231_CR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_232_CR__onD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_CR_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( abstract_join_nat @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_233_CR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% CR_onD
thf(fact_234_CR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_235_CR__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_CR_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( abstract_join_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_236_CR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ) ).
% CR_onE
thf(fact_237_CR__on__def,axiom,
( abstra8448919418672941150rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% CR_on_def
thf(fact_238_rstep__converse__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) )
= ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ).
% rstep_converse_dist
thf(fact_239_in__rtrancl__UnI,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ S4 ) ) )
=> ( member5869715511025134514rm_a_b @ X @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S4 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_240_join__sym,axiom,
! [S: term_a_b,T: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ S ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% join_sym
thf(fact_241_conversion__inv,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R5 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( abstra1130568886744591910rm_a_b @ R5 ) ) ) ).
% conversion_inv
thf(fact_242_conversion__refl,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ).
% conversion_refl
thf(fact_243_symcl__srsteps__conversion,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% symcl_srsteps_conversion
thf(fact_244_asym__onD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( asym_o1443355727950249531rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ Y @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_245_asym__onD,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( asym_on_nat @ A @ R )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_246_asym__onD,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( asym_on_term_a_b @ A @ R )
=> ( ( member_term_a_b @ X @ A )
=> ( ( member_term_a_b @ Y @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_247_asym__on__def,axiom,
( asym_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).
% asym_on_def
thf(fact_248_CR__join__left__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_249_CR__join__left__I,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( abstract_join_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( abstract_join_nat @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_250_CR__join__left__I,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ).
% CR_join_left_I
thf(fact_251_CR__join__right__I,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_252_CR__join__right__I,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( abstract_join_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( abstract_join_nat @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_253_CR__join__right__I,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ).
% CR_join_right_I
thf(fact_254_partially__localize__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X4: term_a_b,Y5: term_a_b,Z3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ Z3 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_255_partially__localize__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
= ( ! [X4: nat,Y5: nat,Z3: nat] :
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z3 ) @ ( transi2905341329935302413cl_nat @ R ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ ( abstract_join_nat @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_256_partially__localize__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b,Y5: produc357393685978478089rm_a_b,Z3: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y5 ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Z3 ) @ ( transi2615809358984392588rm_a_b @ R ) ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ Z3 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ).
% partially_localize_CR
thf(fact_257_CR__divergence__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_258_CR__divergence__imp__join,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( abstract_join_nat @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_259_CR__divergence__imp__join,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ Z2 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ).
% CR_divergence_imp_join
thf(fact_260_UNC__E,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra8480497335821202873rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_261_UNC__E,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( abstract_UNC_nat @ A )
=> ( ( A2 != B2 )
=> ( ( member_nat @ A2 @ ( abstract_NF_nat @ A ) )
=> ( ( member_nat @ B2 @ ( abstract_NF_nat @ A ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra2375765306220057826on_nat @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_262_UNC__E,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ A )
=> ( ( A2 != B2 )
=> ( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B2 @ ( abstract_NF_term_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) ) ) ) ) ) ).
% UNC_E
thf(fact_263_UNC__I,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ B @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ ( abstra6125656530262095991rm_a_b @ A ) )
=> ( A4 = B ) ) ) )
=> ( abstra8480497335821202873rm_a_b @ A ) ) ).
% UNC_I
thf(fact_264_UNC__I,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [A4: nat,B: nat] :
( ( member_nat @ A4 @ ( abstract_NF_nat @ A ) )
=> ( ( member_nat @ B @ ( abstract_NF_nat @ A ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ ( abstra2375765306220057826on_nat @ A ) )
=> ( A4 = B ) ) ) )
=> ( abstract_UNC_nat @ A ) ) ).
% UNC_I
thf(fact_265_UNC__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B: term_a_b] :
( ( member_term_a_b @ A4 @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member_term_a_b @ B @ ( abstract_NF_term_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( A4 = B ) ) ) )
=> ( abstra4649771476585718372rm_a_b @ A ) ) ).
% UNC_I
thf(fact_266_UNC__def,axiom,
( abstra8480497335821202873rm_a_b
= ( ^ [A6: set_Pr2972776593051762503rm_a_b] :
! [A5: produc357393685978478089rm_a_b,B4: produc357393685978478089rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A5 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ B4 @ ( abstra1902783451776770197rm_a_b @ A6 ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A5 @ B4 ) @ ( abstra6125656530262095991rm_a_b @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_267_UNC__def,axiom,
( abstract_UNC_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat] :
! [A5: nat,B4: nat] :
( ( ( member_nat @ A5 @ ( abstract_NF_nat @ A6 ) )
& ( member_nat @ B4 @ ( abstract_NF_nat @ A6 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( abstra2375765306220057826on_nat @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_268_UNC__def,axiom,
( abstra4649771476585718372rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b] :
! [A5: term_a_b,B4: term_a_b] :
( ( ( member_term_a_b @ A5 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member_term_a_b @ B4 @ ( abstract_NF_term_a_b @ A6 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ A6 ) ) )
=> ( A5 = B4 ) ) ) ) ).
% UNC_def
thf(fact_269_CE__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) )
=> ( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ S5 @ ( conver7608559976937845093rm_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) )
=> ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b ) ) ) ).
% CE_rrstep_intro
thf(fact_270_CE__to__rrstep_I1_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) ) ) ).
% CE_to_rrstep(1)
thf(fact_271_CE__to__rrstep_I2_J,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit660295756789745446rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ S5 @ ( conver7608559976937845093rm_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( abstra1130568886744591910rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% CE_to_rrstep(2)
thf(fact_272_CR__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ R3 @ S6 @ T3 ) )
=> ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% CR_rrstep_intro
thf(fact_273_tranclD,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ) ).
% tranclD
thf(fact_274_rtranclD,axiom,
! [A2: term_a_b,B2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ( ( A2 = B2 )
| ( ( A2 != B2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_275_tranclD2,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) )
=> ? [Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_276_trancl__into__rtrancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_277_rtrancl__eq__or__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_278_rtrancl__into__trancl1,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_279_rtrancl__into__trancl2,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_280_rtrancl__trancl__trancl,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_281_trancl__rtrancl__trancl,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_282_rtrancl__converseD,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ).
% rtrancl_converseD
thf(fact_283_rtrancl__converseI,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_284_peak__iff,axiom,
! [X: term_a_b,Y: term_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ B6 ) )
= ( ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ X ) @ A )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U3 @ Y ) @ B6 ) ) ) ) ).
% peak_iff
thf(fact_285_NF__no__trancl__step,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ A ) )
=> ! [B5: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B5 ) @ ( transi9211502839322181930rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_286_NF__no__trancl__step,axiom,
! [A2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ A ) )
=> ! [B5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B5 ) @ ( transi6264000038957366511cl_nat @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_287_NF__no__trancl__step,axiom,
! [A2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ A ) )
=> ! [B5: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B5 ) @ ( transi7922773638565587891rm_a_b @ A ) ) ) ).
% NF_no_trancl_step
thf(fact_288_rtrancl__Un__separatorE,axiom,
! [A2: term_a_b,B2: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ Q )
=> ( X2 = Y3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_289_rtrancl__Un__separator__converseE,axiom,
! [A2: term_a_b,B2: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ Q )
=> ( Y3 = X2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_290_sig__step__converse__rstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ ( rstep_a_b @ R3 ) ) ) ) ).
% sig_step_converse_rstep
thf(fact_291_srstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% srstep_converse_dist
thf(fact_292_CR__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ R3 @ S @ T ) ) ) ).
% CR_to_rrstep
thf(fact_293_UNC__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R3 @ S @ T ) ) ) ).
% UNC_to_rrstep
thf(fact_294_UNC__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( srstep7844470518422762656ep_a_b @ F2 @ ( sup_su6776935440552674877rm_a_b @ R3 @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R3 @ S6 @ T3 ) )
=> ( abstra4649771476585718372rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ).
% UNC_rrstep_intro
thf(fact_295_srrstep__converse__dist,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( conver7608559976937845093rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R3 ) ) )
= ( sig_step_a_b @ F2 @ ( rrstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) ).
% srrstep_converse_dist
thf(fact_296_joinD,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ? [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinD
thf(fact_297_joinE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ~ ! [C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ).
% joinE
thf(fact_298_join__rtrancl__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% join_rtrancl_join
thf(fact_299_rtrancl__join__join,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b,C: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ) ).
% rtrancl_join_join
thf(fact_300_CR__Aux,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ R3 @ S6 @ T3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ R3 @ S @ T ) ) ) ).
% CR_Aux
thf(fact_301_CR__NF__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,T: term_a_b,U: term_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra1130568886744591910rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ T ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_302_CR__NF__conv,axiom,
! [R: set_Pr1261947904930325089at_nat,T: nat,U: nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( ( member_nat @ T @ ( abstract_NF_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ T ) @ ( abstra2375765306220057826on_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ T ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_303_CR__NF__conv,axiom,
! [R: set_Pr2972776593051762503rm_a_b,T: produc357393685978478089rm_a_b,U: produc357393685978478089rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra6125656530262095991rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ U @ T ) @ ( abstra1496011390211842225rm_a_b @ R ) ) ) ) ) ).
% CR_NF_conv
thf(fact_304_UNF__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [T3: term_a_b,U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) ) )
=> ( rewrit983855061078472681dp_a_b @ F2 @ R3 @ T3 @ U2 ) )
=> ( abstra3619855735679963878rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% UNF_rrstep_intro
thf(fact_305_NFP__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) ) ) )
=> ( rewrit232733863534887208dp_a_b @ F2 @ R3 @ S6 @ T3 ) )
=> ( rewrit6304709981509430748rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ top_top_set_term_a_b ) ) ).
% NFP_rrstep_intro
thf(fact_306_join__NF__imp__eq,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ X @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_307_join__NF__imp__eq,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( abstract_join_nat @ R ) )
=> ( ( member_nat @ X @ ( abstract_NF_nat @ R ) )
=> ( ( member_nat @ Y @ ( abstract_NF_nat @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_308_join__NF__imp__eq,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member_term_a_b @ X @ ( abstract_NF_term_a_b @ R ) )
=> ( ( member_term_a_b @ Y @ ( abstract_NF_term_a_b @ R ) )
=> ( X = Y ) ) ) ) ).
% join_NF_imp_eq
thf(fact_309_NFP__lift__to__conversion,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( rewrit6304709981509430748rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ ( conver7608559976937845093rm_a_b @ R ) ) ) )
=> ( ( member_term_a_b @ T @ ( abstract_NF_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_310_NFP__lift__to__conversion,axiom,
! [R: set_Pr1261947904930325089at_nat,S: nat,T: nat] :
( ( rewriting_NFP_on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ R @ ( converse_nat_nat @ R ) ) ) )
=> ( ( member_nat @ T @ ( abstract_NF_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_311_NFP__lift__to__conversion,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: produc357393685978478089rm_a_b,T: produc357393685978478089rm_a_b] :
( ( rewrit2138790387785322561rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R @ ( conver1575690562132235771rm_a_b @ R ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ T @ ( abstra1902783451776770197rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ S @ T ) @ ( transi2615809358984392588rm_a_b @ R ) ) ) ) ) ).
% NFP_lift_to_conversion
thf(fact_312_asymD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( asym_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) ) ) ).
% asymD
thf(fact_313_asymD,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ).
% asymD
thf(fact_314_asymD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ X ) @ R ) ) ) ).
% asymD
thf(fact_315_asym__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( asym_on_term_a_b @ top_top_set_term_a_b @ R )
= ( ! [X4: term_a_b,Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_316_asym__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
= ( ! [X4: nat,Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_317_asym__iff,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R )
= ( ! [X4: produc357393685978478089rm_a_b,Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X4 @ Y5 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_318_CR__imp__UNF,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra3619855735679963878rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% CR_imp_UNF
thf(fact_319_CR__imp__UNF,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
=> ( abstract_UNF_on_nat @ R @ top_top_set_nat ) ) ).
% CR_imp_UNF
thf(fact_320_CR__imp__UNF,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( abstra1303798319023342519rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% CR_imp_UNF
thf(fact_321_firstStep,axiom,
! [L2: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( L2
= ( sup_su6776935440552674877rm_a_b @ S4 @ R5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ L2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R5 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R5 ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_322_first__step,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B6 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ C2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z2 ) @ B6 )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B6 ) ) ) ) ) ).
% first_step
thf(fact_323_first__step__O,axiom,
! [C2: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( C2
= ( sup_su6776935440552674877rm_a_b @ A @ B6 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C2 ) @ B6 ) )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ B6 ) ) ) ) ).
% first_step_O
thf(fact_324_right__comp__S,axiom,
! [X: term_a_b,Y: term_a_b,S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_325_CE__on__def,axiom,
( rewrit660295756789745446rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ R4 ) )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ ( abstra1130568886744591910rm_a_b @ S2 ) ) ) ) ) ) ).
% CE_on_def
thf(fact_326_commute__rrstep__intro,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S5 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ S5 @ S6 @ T3 ) )
=> ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) ).
% commute_rrstep_intro
thf(fact_327_commute__to__rrstep,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( abstra455195320138238452rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ ( conver7608559976937845093rm_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ ( conver7608559976937845093rm_a_b @ R3 ) ) ) ) @ ( srstep7844470518422762656ep_a_b @ F2 @ S5 ) ) ) )
=> ( rewrit8865997592796547114dp_a_b @ F2 @ R3 @ S5 @ S @ T ) ) ) ).
% commute_to_rrstep
thf(fact_328_boolean__algebra_Odisj__one__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_329_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_330_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% boolean_algebra.disj_one_right
thf(fact_331_boolean__algebra_Odisj__one__left,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ X )
= top_top_set_term_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_332_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_333_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= top_to1314267278846557113rm_a_b ) ).
% boolean_algebra.disj_one_left
thf(fact_334_sup__top__right,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% sup_top_right
thf(fact_335_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_336_sup__top__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% sup_top_right
thf(fact_337_sup__top__left,axiom,
! [X: set_term_a_b] :
( ( sup_sup_set_term_a_b @ top_top_set_term_a_b @ X )
= top_top_set_term_a_b ) ).
% sup_top_left
thf(fact_338_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_339_sup__top__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( sup_su6776935440552674877rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= top_to1314267278846557113rm_a_b ) ).
% sup_top_left
thf(fact_340_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( abstra7162307272211697846rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( abstra1130568886744591910rm_a_b @ R ) )
= ( ! [U3: term_a_b,V2: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ U3 ) @ ( abstra2972651231392952684rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ V2 ) @ ( abstra2972651231392952684rm_a_b @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_341_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( abstra8086869556436074354on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( abstra2375765306220057826on_nat @ R ) )
= ( ! [U3: nat,V2: nat] :
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ U3 ) @ ( abstra7312431243108365096ty_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ V2 ) @ ( abstra7312431243108365096ty_nat @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_342_semi__complete__imp__conversionIff__same__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( abstra5399326240126609895rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( abstra6125656530262095991rm_a_b @ R ) )
= ( ! [U3: produc357393685978478089rm_a_b,V2: produc357393685978478089rm_a_b] :
( ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ U3 ) @ ( abstra1496011390211842225rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ V2 ) @ ( abstra1496011390211842225rm_a_b @ R ) ) )
=> ( U3 = V2 ) ) ) ) ) ).
% semi_complete_imp_conversionIff_same_NF
thf(fact_343_weak__diamond__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra5190899124789103926rm_a_b @ R )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% weak_diamond_imp_CR
thf(fact_344_weak__diamond__imp__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstra3280969490040617970nd_nat @ R )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ).
% weak_diamond_imp_CR
thf(fact_345_weak__diamond__imp__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra4281209738456927591rm_a_b @ R )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% weak_diamond_imp_CR
thf(fact_346_trancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
=> ( ! [A4: term_a_b,B: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B ) ) @ R )
=> ( P2 @ A4 @ B ) )
=> ( ! [A4: term_a_b,B: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A4 @ B ) ) @ ( transi9211502839322181930rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A4 @ B )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_347_CR__iff__self__commute,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( abstra455195320138238452rm_a_b @ R @ R ) ) ).
% CR_iff_self_commute
thf(fact_348_CR__iff__self__commute,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
= ( abstract_commute_nat @ R @ R ) ) ).
% CR_iff_self_commute
thf(fact_349_CR__iff__self__commute,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
= ( abstra5145658890658669097rm_a_b @ R @ R ) ) ).
% CR_iff_self_commute
thf(fact_350_trancl__map,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,S: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X ) @ ( F3 @ Y ) ) @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_map
thf(fact_351_semi__confluence__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_352_semi__confluence__imp__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( converse_nat_nat @ R ) @ ( transi2905341329935302413cl_nat @ R ) ) @ ( abstract_join_nat @ R ) )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ).
% semi_confluence_imp_CR
thf(fact_353_semi__confluence__imp__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) @ ( transi2615809358984392588rm_a_b @ R ) ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% semi_confluence_imp_CR
thf(fact_354_WCR__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b,B: produc357393685978478089rm_a_b,C4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ C4 ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B @ C4 ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) )
=> ( abstra875157153669124638rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_355_WCR__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat,B: nat,C4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ C4 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C4 ) @ ( abstract_join_nat @ R ) ) ) ) )
=> ( abstract_WCR_on_nat @ R @ A ) ) ).
% WCR_onI
thf(fact_356_WCR__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b,B: term_a_b,C4: term_a_b] :
( ( member_term_a_b @ A4 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ C4 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) )
=> ( abstra5026849107050860991rm_a_b @ R @ A ) ) ).
% WCR_onI
thf(fact_357_strong__confluence__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strong_confluence_imp_CR
thf(fact_358_strong__confluence__imp__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstra1637537658730489263on_nat @ R @ top_top_set_nat )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ).
% strong_confluence_imp_CR
thf(fact_359_strong__confluence__imp__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% strong_confluence_imp_CR
thf(fact_360_comp__rtrancl__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R5 @ S4 ) ) @ S4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ S4 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_361_CR__iff__meet__subset__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1093061187967292567rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_362_CR__iff__meet__subset__join,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
= ( ord_le3146513528884898305at_nat @ ( abstract_meet_nat @ R ) @ ( abstract_join_nat @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_363_CR__iff__meet__subset__join,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
= ( ord_le2593958535671617255rm_a_b @ ( abstra8296971427885378886rm_a_b @ R ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ).
% CR_iff_meet_subset_join
thf(fact_364_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_365_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_366_subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( member5869715511025134514rm_a_b @ X2 @ B6 ) )
=> ( ord_le118470702582115849rm_a_b @ A @ B6 ) ) ).
% subsetI
thf(fact_367_subsetI,axiom,
! [A: set_nat,B6: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B6 ) )
=> ( ord_less_eq_set_nat @ A @ B6 ) ) ).
% subsetI
thf(fact_368_subsetI,axiom,
! [A: set_term_a_b,B6: set_term_a_b] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( member_term_a_b @ X2 @ B6 ) )
=> ( ord_le2705286416250468010rm_a_b @ A @ B6 ) ) ).
% subsetI
thf(fact_369_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_370_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_371_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_372_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_373_NF__anti__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B6 )
=> ( ord_le2705286416250468010rm_a_b @ ( abstract_NF_term_a_b @ B6 ) @ ( abstract_NF_term_a_b @ A ) ) ) ).
% NF_anti_mono
thf(fact_374_srsteps__with__root__step__sig__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( srstep7844470518422762656ep_a_b @ F2 @ R3 ) @ ( srstep7844470518422762656ep_a_b @ G2 @ R3 ) ) ) ).
% srsteps_with_root_step_sig_mono
thf(fact_375_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_376_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_377_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_378_ord__eq__le__subst,axiom,
! [A2: nat,F3: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_379_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_380_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_381_order__subst2,axiom,
! [A2: nat,B2: nat,F3: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F3 @ B2 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_382_order__subst1,axiom,
! [A2: nat,F3: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_383_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_384_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_385_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B7: set_Pr4386577575007340137rm_a_b] :
! [T2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T2 @ A6 )
=> ( member5869715511025134514rm_a_b @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_386_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_387_subset__iff,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B7: set_term_a_b] :
! [T2: term_a_b] :
( ( member_term_a_b @ T2 @ A6 )
=> ( member_term_a_b @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_388_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B7: set_Pr4386577575007340137rm_a_b] :
! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A6 )
=> ( member5869715511025134514rm_a_b @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_389_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_390_subset__eq,axiom,
( ord_le2705286416250468010rm_a_b
= ( ^ [A6: set_term_a_b,B7: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ( member_term_a_b @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_391_subsetD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,C: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B6 )
=> ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% subsetD
thf(fact_392_subsetD,axiom,
! [A: set_nat,B6: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B6 ) ) ) ).
% subsetD
thf(fact_393_subsetD,axiom,
! [A: set_term_a_b,B6: set_term_a_b,C: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B6 )
=> ( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B6 ) ) ) ).
% subsetD
thf(fact_394_in__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ B6 )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ X @ B6 ) ) ) ).
% in_mono
thf(fact_395_in__mono,axiom,
! [A: set_nat,B6: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B6 ) ) ) ).
% in_mono
thf(fact_396_in__mono,axiom,
! [A: set_term_a_b,B6: set_term_a_b,X: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ B6 )
=> ( ( member_term_a_b @ X @ A )
=> ( member_term_a_b @ X @ B6 ) ) ) ).
% in_mono
thf(fact_397_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_398_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_399_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_400_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( P2 @ A4 @ B ) )
=> ( ! [A4: nat,B: nat] :
( ( P2 @ B @ A4 )
=> ( P2 @ A4 @ B ) )
=> ( P2 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_401_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_402_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_403_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_404_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_405_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_406_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_407_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_408_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_409_sig__step__mono,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R3 ) @ ( sig_step_a_b @ G2 @ R3 ) ) ) ).
% sig_step_mono
thf(fact_410_srstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R3 ) ) ) ) ).
% srstep_monp
thf(fact_411_subrelI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R @ S ) ) ).
% subrelI
thf(fact_412_top_Oextremum__uniqueI,axiom,
! [A2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A2 )
=> ( A2 = top_top_set_term_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_413_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_414_top_Oextremum__uniqueI,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
=> ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_uniqueI
thf(fact_415_top_Oextremum__unique,axiom,
! [A2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ top_top_set_term_a_b @ A2 )
= ( A2 = top_top_set_term_a_b ) ) ).
% top.extremum_unique
thf(fact_416_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_417_top_Oextremum__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
= ( A2 = top_to1314267278846557113rm_a_b ) ) ).
% top.extremum_unique
thf(fact_418_top__greatest,axiom,
! [A2: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A2 @ top_top_set_term_a_b ) ).
% top_greatest
thf(fact_419_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_420_top__greatest,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A2 @ top_to1314267278846557113rm_a_b ) ).
% top_greatest
thf(fact_421_subset__UNIV,axiom,
! [A: set_term_a_b] : ( ord_le2705286416250468010rm_a_b @ A @ top_top_set_term_a_b ) ).
% subset_UNIV
thf(fact_422_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_423_subset__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ).
% subset_UNIV
thf(fact_424_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_425_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_426_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_427_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_428_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_429_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_430_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_431_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_432_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_433_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_434_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_435_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( sup_sup_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_436_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_437_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_438_sup__unique,axiom,
! [F3: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F3 @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F3 @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( F3 @ Y3 @ Z ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_439_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_440_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_441_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_442_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_443_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_444_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_445_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_446_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_447_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_448_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_449_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_450_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_451_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_452_srrstep__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R3 ) ) @ ( sig_step_a_b @ G2 @ ( rrstep_a_b @ R3 ) ) ) ) ).
% srrstep_monp
thf(fact_453_cSup__eq__maximum,axiom,
! [Z2: nat,X3: set_nat] :
( ( member_nat @ Z2 @ X3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X3 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X3 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_454_trancl__mono,axiom,
! [P: produc357393685978478089rm_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7922773638565587891rm_a_b @ S ) ) ) ) ).
% trancl_mono
thf(fact_455_sig__step__mono2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,L3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le118470702582115849rm_a_b @ R3 @ L3 )
=> ( ord_le118470702582115849rm_a_b @ ( sig_step_a_b @ F2 @ R3 ) @ ( sig_step_a_b @ F2 @ L3 ) ) ) ).
% sig_step_mono2
thf(fact_456_srsteps__eq__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_eq_monp
thf(fact_457_srsteps__monp,axiom,
! [F2: set_Pr4934435412358123699_a_nat,G2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ F2 @ G2 )
=> ( ord_le118470702582115849rm_a_b @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ G2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srsteps_monp
thf(fact_458_not__UNIV__le__Iic,axiom,
! [H2: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_le_Iic
thf(fact_459_rrstep__rstep__mono,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( rrstep_a_b @ R3 ) @ ( rstep_a_b @ R3 ) ) ).
% rrstep_rstep_mono
thf(fact_460_WCR__alt__def,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra5026849107050860991rm_a_b @ A @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( conver7608559976937845093rm_a_b @ A ) @ A ) @ ( abstra4096080454567261402rm_a_b @ A ) ) ) ).
% WCR_alt_def
thf(fact_461_WCR__alt__def,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( abstract_WCR_on_nat @ A @ top_top_set_nat )
= ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( converse_nat_nat @ A ) @ A ) @ ( abstract_join_nat @ A ) ) ) ).
% WCR_alt_def
thf(fact_462_WCR__alt__def,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra875157153669124638rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( conver1575690562132235771rm_a_b @ A ) @ A ) @ ( abstra4889682940153293507rm_a_b @ A ) ) ) ).
% WCR_alt_def
thf(fact_463_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S4 ) ) ) ).
% non_strict_into_strict
thf(fact_464_WCR__on__def,axiom,
( abstra5026849107050860991rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ R4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ C3 ) @ ( abstra4096080454567261402rm_a_b @ R4 ) ) ) ) ) ) ).
% WCR_on_def
thf(fact_465_WCR__onE,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_466_WCR__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_WCR_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( abstract_join_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_467_WCR__onE,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R ) ) ) ) ) ).
% WCR_onE
thf(fact_468_WCR__onD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,C: produc357393685978478089rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ C ) @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ C ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_469_WCR__onD,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat,B2: nat,C: nat] :
( ( abstract_WCR_on_nat @ R @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ ( abstract_join_nat @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_470_WCR__onD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,A2: term_a_b,B2: term_a_b,C: term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ A )
=> ( ( member_term_a_b @ A2 @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ C ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ) ) ) ).
% WCR_onD
thf(fact_471_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_472_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b )
= ( ord_le118470702582115849rm_a_b @ ( abstra1130568886744591910rm_a_b @ R ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_473_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_CR_on_nat @ R @ top_top_set_nat )
= ( ord_le3146513528884898305at_nat @ ( abstra2375765306220057826on_nat @ R ) @ ( abstract_join_nat @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_474_CR__iff__conversion__imp__join,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b )
= ( ord_le2593958535671617255rm_a_b @ ( abstra6125656530262095991rm_a_b @ R ) @ ( abstra4889682940153293507rm_a_b @ R ) ) ) ).
% CR_iff_conversion_imp_join
thf(fact_475_diamond__imp__CR_H,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ S )
=> ( ( ord_le118470702582115849rm_a_b @ R @ S )
=> ( ( ord_le118470702582115849rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_476_diamond__imp__CR_H,axiom,
! [S: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
( ( abstract_diamond_nat @ S )
=> ( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ( ord_le3146513528884898305at_nat @ S @ ( transi2905341329935302413cl_nat @ R ) )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ) ) ).
% diamond_imp_CR'
thf(fact_477_diamond__imp__CR_H,axiom,
! [S: set_Pr2972776593051762503rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7467186385360731951rm_a_b @ S )
=> ( ( ord_le2593958535671617255rm_a_b @ R @ S )
=> ( ( ord_le2593958535671617255rm_a_b @ S @ ( transi2615809358984392588rm_a_b @ R ) )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ) ) ).
% diamond_imp_CR'
thf(fact_478_diamond__imp__CR,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra8151087076625346926rm_a_b @ R )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% diamond_imp_CR
thf(fact_479_diamond__imp__CR,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_diamond_nat @ R )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ).
% diamond_imp_CR
thf(fact_480_diamond__imp__CR,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra7467186385360731951rm_a_b @ R )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% diamond_imp_CR
thf(fact_481_strongly__confluentI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b,Z: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Z ) @ R )
=> ? [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ U4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z @ U4 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) )
=> ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b ) ) ).
% strongly_confluentI
thf(fact_482_strongly__confluentI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z ) @ R )
=> ? [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ U4 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ U4 ) @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) ) ) ) )
=> ( abstra1637537658730489263on_nat @ R @ top_top_set_nat ) ) ).
% strongly_confluentI
thf(fact_483_strongly__confluentI,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b,Z: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Z ) @ R )
=> ? [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ U4 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z @ U4 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) )
=> ( abstra837640964651051114rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ).
% strongly_confluentI
thf(fact_484_sig__step__rsteps__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
= ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% sig_step_rsteps_dist
thf(fact_485_all__ctxt__closed__relcomp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ R3 )
=> ( ( S6 != T3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S6 ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T3 ) @ F2 ) ) ) )
=> ( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ S5 )
=> ( ( S6 != T3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S6 ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T3 ) @ F2 ) ) ) )
=> ( ( terms_5226143800768910156ed_a_b @ F2 @ R3 )
=> ( ( terms_5226143800768910156ed_a_b @ F2 @ S5 )
=> ( terms_5226143800768910156ed_a_b @ F2 @ ( relcom370159955682700863rm_a_b @ R3 @ S5 ) ) ) ) ) ) ).
% all_ctxt_closed_relcomp
thf(fact_486_srstepsD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srstepsD
thf(fact_487_IdI,axiom,
! [A2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ id_term_a_b ) ).
% IdI
thf(fact_488_pair__in__Id__conv,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_489_sig__stepI,axiom,
! [S: term_a_b,F2: set_Pr4934435412358123699_a_nat,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R3 ) ) ) ) ) ).
% sig_stepI
thf(fact_490_rstep__srstepI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_srstepI
thf(fact_491_rsteps__eq__srsteps__eqI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% rsteps_eq_srsteps_eqI
thf(fact_492_rsteps__srstepsI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ) ).
% rsteps_srstepsI
thf(fact_493_rsteps__eq__relcomp__srsteps__eq__relcompI,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ S5 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ R3 ) ) @ ( transi7742714808557438673rm_a_b @ ( rstep_a_b @ S5 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ) ) ) ).
% rsteps_eq_relcomp_srsteps_eq_relcompI
thf(fact_494_IdE,axiom,
! [P: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ id_term_a_b )
=> ~ ! [X2: term_a_b] :
( P
!= ( produc7020197800436672577rm_a_b @ X2 @ X2 ) ) ) ).
% IdE
thf(fact_495_sig__stepE,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ R3 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ R3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ~ ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ) ).
% sig_stepE
thf(fact_496_all__ctxt__closed__sig__reflE,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,T: term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F2 @ R3 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ R3 ) ) ) ).
% all_ctxt_closed_sig_reflE
thf(fact_497_srstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srstepD
thf(fact_498_sig__step__rsteps__eq__dist,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( sup_su6776935440552674877rm_a_b @ ( sig_step_a_b @ F2 @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) ) @ id_term_a_b )
= ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% sig_step_rsteps_eq_dist
thf(fact_499_srrstepD,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rrstep_a_b @ R3 ) ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R3 ) )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% srrstepD
thf(fact_500_strongly__confluent__on__E11,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b] :
( ( abstra837640964651051114rm_a_b @ R @ A )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ R )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_501_strongly__confluent__on__E11,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat,X: nat,Y: nat,Z2: nat] :
( ( abstra1637537658730489263on_nat @ R @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R )
=> ? [U2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ U2 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ U2 ) @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_502_strongly__confluent__on__E11,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b] :
( ( abstra5664096641628173427rm_a_b @ R @ A )
=> ( ( member_term_a_b @ X @ A )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ R )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_E11
thf(fact_503_strongly__confluent__on__def,axiom,
( abstra5664096641628173427rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b,Z3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Z3 ) @ R4 ) )
=> ? [U3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ U3 ) @ ( transi7742714808557438673rm_a_b @ R4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z3 @ U3 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) ) ) ) ) ) ) ).
% strongly_confluent_on_def
thf(fact_504_sigstep__trancl__funas_I1_J,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 ) ) ) ).
% sigstep_trancl_funas(1)
thf(fact_505_sigstep__trancl__funas_I2_J,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,S5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) )
=> ( ( S != T )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 ) ) ) ).
% sigstep_trancl_funas(2)
thf(fact_506_SCR__on__def,axiom,
( rewrit4966517460798004134rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b,A6: set_term_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [B4: term_a_b,C3: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ B4 ) @ R4 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ C3 ) @ R4 ) )
=> ? [D2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B4 @ D2 ) @ ( sup_su6776935440552674877rm_a_b @ R4 @ id_term_a_b ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ C3 @ D2 ) @ ( transi7742714808557438673rm_a_b @ R4 ) ) ) ) ) ) ) ).
% SCR_on_def
thf(fact_507_rstep__trancl__sig__step__l,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_l
thf(fact_508_rstep__trancl__sig__step__r,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( rstep_a_b @ R3 ) ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ) ).
% rstep_trancl_sig_step_r
thf(fact_509_IdD,axiom,
! [A2: term_a_b,B2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ id_term_a_b )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_510_rstep__term__to__sig__r,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T ) ) @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% rstep_term_to_sig_r
thf(fact_511_rstep__term__to__sig__l,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( terms_7988297476397195622_a_b_b @ R3 ) @ F2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) @ T ) @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% rstep_term_to_sig_l
thf(fact_512_srsteps__eq__subst__relcomp__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) )
=> ( ! [X2: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X2 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ S5 ) ) ) ) ) ) ) ).
% srsteps_eq_subst_relcomp_closed
thf(fact_513_strongly__confluent__E1n,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,Z2: term_a_b,N: nat] :
( ( abstra5664096641628173427rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Z2 ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ? [U2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ U2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ U2 ) @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_514_strongly__confluent__E1n,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat,N: nat] :
( ( abstra1637537658730489263on_nat @ R @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( compow1820118003163928468at_nat @ N @ R ) )
=> ? [U2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ U2 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ U2 ) @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_515_strongly__confluent__E1n,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,Z2: produc357393685978478089rm_a_b,N: nat] :
( ( abstra837640964651051114rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Z2 ) @ ( compow1649094434264376826rm_a_b @ N @ R ) )
=> ? [U2: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y @ U2 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Z2 @ U2 ) @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ).
% strongly_confluent_E1n
thf(fact_516_srsteps__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) )
=> ( ! [X2: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X2 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7922773638565587891rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_subst_closed
thf(fact_517_Diff__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ~ ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_518_Diff__iff,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B6 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_519_Diff__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) )
= ( ( member_term_a_b @ C @ A )
& ~ ( member_term_a_b @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_520_DiffI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ C @ B6 )
=> ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) ) ) ) ).
% DiffI
thf(fact_521_DiffI,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B6 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B6 ) ) ) ) ).
% DiffI
thf(fact_522_DiffI,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( ~ ( member_term_a_b @ C @ B6 )
=> ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) ) ) ) ).
% DiffI
thf(fact_523_rstep__substI,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rstep_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( rstep_a_b @ R3 ) ) ) ).
% rstep_substI
thf(fact_524_relpow__refl__mono,axiom,
! [Rel: set_Pr4386577575007340137rm_a_b,M: nat,N: nat,A2: term_a_b,B2: term_a_b] :
( ! [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ Rel )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ M @ Rel ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( compow4057154403645558940rm_a_b @ N @ Rel ) ) ) ) ) ).
% relpow_refl_mono
thf(fact_525_relpow__Suc__D2_H,axiom,
! [N: nat,R5: set_Pr4386577575007340137rm_a_b,X6: term_a_b,Y7: term_a_b,Z5: term_a_b] :
( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y7 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z5 ) @ R5 ) )
=> ? [W: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ W ) @ R5 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ W @ Z5 ) @ ( compow4057154403645558940rm_a_b @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_526_relpow__image,axiom,
! [R: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b,R7: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b,N: nat] :
( ! [S6: term_a_b,T3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S6 @ T3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S6 ) @ ( F3 @ T3 ) ) @ R7 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( compow4057154403645558940rm_a_b @ N @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ S ) @ ( F3 @ T ) ) @ ( compow4057154403645558940rm_a_b @ N @ R7 ) ) ) ) ).
% relpow_image
thf(fact_527_rtrancl__imp__relpow,axiom,
! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) )
=> ? [N2: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N2 @ R5 ) ) ) ).
% rtrancl_imp_relpow
thf(fact_528_relpow__imp__rtrancl,axiom,
! [P: produc357393685978478089rm_a_b,N: nat,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N @ R5 ) )
=> ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) ) ) ).
% relpow_imp_rtrancl
thf(fact_529_rtrancl__power,axiom,
! [P: produc357393685978478089rm_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ P @ ( transi7742714808557438673rm_a_b @ R5 ) )
= ( ? [N3: nat] : ( member5869715511025134514rm_a_b @ P @ ( compow4057154403645558940rm_a_b @ N3 @ R5 ) ) ) ) ).
% rtrancl_power
thf(fact_530_DiffD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) )
=> ~ ( member5869715511025134514rm_a_b @ C @ B6 ) ) ).
% DiffD2
thf(fact_531_DiffD2,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B6 ) )
=> ~ ( member_nat @ C @ B6 ) ) ).
% DiffD2
thf(fact_532_DiffD2,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) )
=> ~ ( member_term_a_b @ C @ B6 ) ) ).
% DiffD2
thf(fact_533_DiffD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_534_DiffD1,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B6 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_535_DiffD1,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) )
=> ( member_term_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_536_DiffE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% DiffE
thf(fact_537_DiffE,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B6 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B6 ) ) ) ).
% DiffE
thf(fact_538_DiffE,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) )
=> ~ ( ( member_term_a_b @ C @ A )
=> ( member_term_a_b @ C @ B6 ) ) ) ).
% DiffE
thf(fact_539_rtrancl__len__E,axiom,
! [X: term_a_b,Y: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ! [N2: nat] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N2 @ R ) ) ) ).
% rtrancl_len_E
thf(fact_540_rrstep__subst,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( rrstep_a_b @ R3 ) )
=> ~ ! [L4: term_a_b,R8: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R8 ) @ R3 )
=> ! [Sigma2: b > term_a_b] :
( ( S
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma2 ) )
=> ( T
!= ( subst_7999470309526761004_a_b_b @ R8 @ Sigma2 ) ) ) ) ) ).
% rrstep_subst
thf(fact_541_rrstep_Ointros,axiom,
! [L: term_a_b,R: term_a_b,R3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L @ R ) @ R3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ L @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ R @ Sigma ) ) @ ( rrstep_a_b @ R3 ) ) ) ).
% rrstep.intros
thf(fact_542_rrstep_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R3 ) )
= ( ? [L5: term_a_b,R4: term_a_b,Sigma3: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L5 @ Sigma3 ) )
& ( A22
= ( subst_7999470309526761004_a_b_b @ R4 @ Sigma3 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L5 @ R4 ) @ R3 ) ) ) ) ).
% rrstep.simps
thf(fact_543_rrstep_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( rrstep_a_b @ R3 ) )
=> ~ ! [L4: term_a_b,R8: term_a_b,Sigma2: b > term_a_b] :
( ( A1
= ( subst_7999470309526761004_a_b_b @ L4 @ Sigma2 ) )
=> ( ( A22
= ( subst_7999470309526761004_a_b_b @ R8 @ Sigma2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ L4 @ R8 ) @ R3 ) ) ) ) ).
% rrstep.cases
thf(fact_544_trancl__steps__relpow,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b,N: nat] :
( ( ord_le118470702582115849rm_a_b @ A2 @ ( transi7922773638565587891rm_a_b @ B2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ N @ A2 ) )
=> ? [M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( compow4057154403645558940rm_a_b @ M2 @ B2 ) ) ) ) ) ).
% trancl_steps_relpow
thf(fact_545_rtrancl__diff__decomp,axiom,
! [X: term_a_b,Y: term_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( transi7742714808557438673rm_a_b @ B6 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) @ ( transi7742714808557438673rm_a_b @ A ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_546_srstep__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) )
=> ( ! [X2: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X2 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ).
% srstep_subst_closed
thf(fact_547_srsteps__eq__subst__closed,axiom,
! [S: term_a_b,T: term_a_b,F2: set_Pr4934435412358123699_a_nat,R3: set_Pr4386577575007340137rm_a_b,Sigma: b > term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) )
=> ( ! [X2: b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( Sigma @ X2 ) ) @ F2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( subst_7999470309526761004_a_b_b @ S @ Sigma ) @ ( subst_7999470309526761004_a_b_b @ T @ Sigma ) ) @ ( transi7742714808557438673rm_a_b @ ( sig_step_a_b @ F2 @ ( rstep_a_b @ R3 ) ) ) ) ) ) ).
% srsteps_eq_subst_closed
thf(fact_548_qc__SN__relto__iff,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R @ S ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S @ R ) ) ) )
=> ( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R ) @ ( relcom370159955682700863rm_a_b @ S @ ( transi7742714808557438673rm_a_b @ R ) ) ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_549_qc__SN__relto__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R @ S ) @ ( relcomp_nat_nat_nat @ S @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ S @ R ) ) ) )
=> ( ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( relcomp_nat_nat_nat @ S @ ( transi2905341329935302413cl_nat @ R ) ) ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ S @ top_top_set_nat ) ) ) ).
% qc_SN_relto_iff
thf(fact_550_qc__SN__relto__iff,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ R @ S ) @ ( relcom6159724393391289352rm_a_b @ S @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ S @ R ) ) ) )
=> ( ( abstra2398554102055911763rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) @ ( relcom6159724393391289352rm_a_b @ S @ ( transi2615809358984392588rm_a_b @ R ) ) ) @ top_to1314267278846557113rm_a_b )
= ( abstra2398554102055911763rm_a_b @ S @ top_to1314267278846557113rm_a_b ) ) ) ).
% qc_SN_relto_iff
thf(fact_551_compatible__SN_H,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) @ ( relcom370159955682700863rm_a_b @ S4 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN'
thf(fact_552_compatible__SN_H,axiom,
! [NS: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S4 ) @ S4 )
=> ( ( abstract_SN_on_nat @ S4 @ top_top_set_nat )
=> ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S4 ) ) @ ( relcomp_nat_nat_nat @ S4 @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S4 ) ) ) ) @ top_top_set_nat ) ) ) ).
% compatible_SN'
thf(fact_553_compatible__SN_H,axiom,
! [NS: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] :
( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( abstra2398554102055911763rm_a_b @ S4 @ top_to1314267278846557113rm_a_b )
=> ( abstra2398554102055911763rm_a_b @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ NS @ S4 ) ) @ ( relcom6159724393391289352rm_a_b @ S4 @ ( transi2615809358984392588rm_a_b @ ( sup_su1845815533690797339rm_a_b @ NS @ S4 ) ) ) ) @ top_to1314267278846557113rm_a_b ) ) ) ).
% compatible_SN'
thf(fact_554_IntI,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ A )
=> ( ( member5869715511025134514rm_a_b @ C @ B6 )
=> ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ) ).
% IntI
thf(fact_555_IntI,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B6 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B6 ) ) ) ) ).
% IntI
thf(fact_556_IntI,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ A )
=> ( ( member_term_a_b @ C @ B6 )
=> ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ) ).
% IntI
thf(fact_557_Int__iff,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) )
= ( ( member5869715511025134514rm_a_b @ C @ A )
& ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_558_Int__iff,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B6 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_559_Int__iff,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B6 ) )
= ( ( member_term_a_b @ C @ A )
& ( member_term_a_b @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_560_inf__top_Oright__neutral,axiom,
! [A2: set_term_a_b] :
( ( inf_inf_set_term_a_b @ A2 @ top_top_set_term_a_b )
= A2 ) ).
% inf_top.right_neutral
thf(fact_561_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_562_inf__top_Oright__neutral,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A2 @ top_to1314267278846557113rm_a_b )
= A2 ) ).
% inf_top.right_neutral
thf(fact_563_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( top_top_set_term_a_b
= ( inf_inf_set_term_a_b @ A2 @ B2 ) )
= ( ( A2 = top_top_set_term_a_b )
& ( B2 = top_top_set_term_a_b ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_564_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_565_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( top_to1314267278846557113rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A2 @ B2 ) )
= ( ( A2 = top_to1314267278846557113rm_a_b )
& ( B2 = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_566_inf__top_Oleft__neutral,axiom,
! [A2: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_567_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_568_inf__top_Oleft__neutral,axiom,
! [A2: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_569_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_term_a_b,B2: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A2 @ B2 )
= top_top_set_term_a_b )
= ( ( A2 = top_top_set_term_a_b )
& ( B2 = top_top_set_term_a_b ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_570_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_571_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,B2: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A2 @ B2 )
= top_to1314267278846557113rm_a_b )
= ( ( A2 = top_to1314267278846557113rm_a_b )
& ( B2 = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_572_top__eq__inf__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( top_top_set_term_a_b
= ( inf_inf_set_term_a_b @ X @ Y ) )
= ( ( X = top_top_set_term_a_b )
& ( Y = top_top_set_term_a_b ) ) ) ).
% top_eq_inf_iff
thf(fact_573_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_574_top__eq__inf__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( top_to1314267278846557113rm_a_b
= ( inf_in1693821330460682071rm_a_b @ X @ Y ) )
= ( ( X = top_to1314267278846557113rm_a_b )
& ( Y = top_to1314267278846557113rm_a_b ) ) ) ).
% top_eq_inf_iff
thf(fact_575_inf__eq__top__iff,axiom,
! [X: set_term_a_b,Y: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ X @ Y )
= top_top_set_term_a_b )
= ( ( X = top_top_set_term_a_b )
& ( Y = top_top_set_term_a_b ) ) ) ).
% inf_eq_top_iff
thf(fact_576_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_577_inf__eq__top__iff,axiom,
! [X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ X @ Y )
= top_to1314267278846557113rm_a_b )
= ( ( X = top_to1314267278846557113rm_a_b )
& ( Y = top_to1314267278846557113rm_a_b ) ) ) ).
% inf_eq_top_iff
thf(fact_578_inf__top__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ top_top_set_term_a_b )
= X ) ).
% inf_top_right
thf(fact_579_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_580_inf__top__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= X ) ).
% inf_top_right
thf(fact_581_inf__top__left,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ X )
= X ) ).
% inf_top_left
thf(fact_582_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_583_inf__top__left,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ X )
= X ) ).
% inf_top_left
thf(fact_584_Int__UNIV,axiom,
! [A: set_term_a_b,B6: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A @ B6 )
= top_top_set_term_a_b )
= ( ( A = top_top_set_term_a_b )
& ( B6 = top_top_set_term_a_b ) ) ) ).
% Int_UNIV
thf(fact_585_Int__UNIV,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( inf_inf_set_nat @ A @ B6 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B6 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_586_Int__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B6 )
= top_to1314267278846557113rm_a_b )
= ( ( A = top_to1314267278846557113rm_a_b )
& ( B6 = top_to1314267278846557113rm_a_b ) ) ) ).
% Int_UNIV
thf(fact_587_IntE,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C @ A )
=> ~ ( member5869715511025134514rm_a_b @ C @ B6 ) ) ) ).
% IntE
thf(fact_588_IntE,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B6 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B6 ) ) ) ).
% IntE
thf(fact_589_IntE,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B6 ) )
=> ~ ( ( member_term_a_b @ C @ A )
=> ~ ( member_term_a_b @ C @ B6 ) ) ) ).
% IntE
thf(fact_590_IntD1,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) )
=> ( member5869715511025134514rm_a_b @ C @ A ) ) ).
% IntD1
thf(fact_591_IntD1,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B6 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_592_IntD1,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B6 ) )
=> ( member_term_a_b @ C @ A ) ) ).
% IntD1
thf(fact_593_IntD2,axiom,
! [C: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) )
=> ( member5869715511025134514rm_a_b @ C @ B6 ) ) ).
% IntD2
thf(fact_594_IntD2,axiom,
! [C: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B6 ) )
=> ( member_nat @ C @ B6 ) ) ).
% IntD2
thf(fact_595_IntD2,axiom,
! [C: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ C @ ( inf_inf_set_term_a_b @ A @ B6 ) )
=> ( member_term_a_b @ C @ B6 ) ) ).
% IntD2
thf(fact_596_Int__Collect__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ( ord_le118470702582115849rm_a_b @ A @ B6 )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( inf_in1693821330460682071rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P2 ) ) @ ( inf_in1693821330460682071rm_a_b @ B6 @ ( collec99177395925924084rm_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_597_Int__Collect__mono,axiom,
! [A: set_nat,B6: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B6 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_598_Int__Collect__mono,axiom,
! [A: set_term_a_b,B6: set_term_a_b,P2: term_a_b > $o,Q: term_a_b > $o] :
( ( ord_le2705286416250468010rm_a_b @ A @ B6 )
=> ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le2705286416250468010rm_a_b @ ( inf_inf_set_term_a_b @ A @ ( collect_term_a_b @ P2 ) ) @ ( inf_inf_set_term_a_b @ B6 @ ( collect_term_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_599_Int__UNIV__right,axiom,
! [A: set_term_a_b] :
( ( inf_inf_set_term_a_b @ A @ top_top_set_term_a_b )
= A ) ).
% Int_UNIV_right
thf(fact_600_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_601_Int__UNIV__right,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= A ) ).
% Int_UNIV_right
thf(fact_602_Int__UNIV__left,axiom,
! [B6: set_term_a_b] :
( ( inf_inf_set_term_a_b @ top_top_set_term_a_b @ B6 )
= B6 ) ).
% Int_UNIV_left
thf(fact_603_Int__UNIV__left,axiom,
! [B6: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B6 )
= B6 ) ).
% Int_UNIV_left
thf(fact_604_Int__UNIV__left,axiom,
! [B6: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ top_to1314267278846557113rm_a_b @ B6 )
= B6 ) ).
% Int_UNIV_left
thf(fact_605_boolean__algebra_Oconj__one__right,axiom,
! [X: set_term_a_b] :
( ( inf_inf_set_term_a_b @ X @ top_top_set_term_a_b )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_606_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_607_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Pr4386577575007340137rm_a_b] :
( ( inf_in1693821330460682071rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_608_SN__on__induct,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b,S: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( abstra2398554102055911763rm_a_b @ R5 @ A )
=> ( ( member5869715511025134514rm_a_b @ S @ A )
=> ( ! [T3: produc357393685978478089rm_a_b] :
( ! [U4: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ T3 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T3 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_609_SN__on__induct,axiom,
! [R5: set_Pr1261947904930325089at_nat,A: set_nat,S: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R5 @ A )
=> ( ( member_nat @ S @ A )
=> ( ! [T3: nat] :
( ! [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T3 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T3 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_610_SN__on__induct,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A: set_term_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ A )
=> ( ( member_term_a_b @ S @ A )
=> ( ! [T3: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T3 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_611_SN__on__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ X6 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_612_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M3: nat] :
( ( P2 @ X )
=> ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M2: nat] :
( ( P2 @ M2 )
=> ~ ! [X6: nat] :
( ( P2 @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_613_SN__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ! [A4: term_a_b] :
( ! [B5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ B5 ) @ R )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) )
=> ( P2 @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_614_SN__induct__rule,axiom,
! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A2: nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( ! [A4: nat] :
( ! [B5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B5 ) @ R )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) )
=> ( P2 @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_615_SN__induct__rule,axiom,
! [R: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o,A2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ! [A4: produc357393685978478089rm_a_b] :
( ! [B5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ B5 ) @ R )
=> ( P2 @ B5 ) )
=> ( P2 @ A4 ) )
=> ( P2 @ A2 ) ) ) ).
% SN_induct_rule
thf(fact_616_SN__imp__minimal,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ! [Q2: set_term_a_b] :
( ? [X6: term_a_b] : ( member_term_a_b @ X6 @ Q2 )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ Q2 )
& ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y7 ) @ A )
=> ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_617_SN__imp__minimal,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ! [Q2: set_nat] :
( ? [X6: nat] : ( member_nat @ X6 @ Q2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ Q2 )
& ! [Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y7 ) @ A )
=> ~ ( member_nat @ Y7 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_618_SN__imp__minimal,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ Q2 )
=> ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ Q2 )
& ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y7 ) @ A )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_619_refl__not__SN,axiom,
! [X: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R5 )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% refl_not_SN
thf(fact_620_refl__not__SN,axiom,
! [X: nat,R5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R5 )
=> ~ ( abstract_SN_on_nat @ R5 @ top_top_set_nat ) ) ).
% refl_not_SN
thf(fact_621_refl__not__SN,axiom,
! [X: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X ) @ R5 )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b ) ) ).
% refl_not_SN
thf(fact_622_SN__subset,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,R9: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ R9 @ R5 )
=> ( abstra4720023341729745482rm_a_b @ R9 @ top_top_set_term_a_b ) ) ) ).
% SN_subset
thf(fact_623_SN__subset,axiom,
! [R5: set_Pr1261947904930325089at_nat,R9: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R5 @ top_top_set_nat )
=> ( ( ord_le3146513528884898305at_nat @ R9 @ R5 )
=> ( abstract_SN_on_nat @ R9 @ top_top_set_nat ) ) ) ).
% SN_subset
thf(fact_624_SN__subset,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,R9: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b )
=> ( ( ord_le2593958535671617255rm_a_b @ R9 @ R5 )
=> ( abstra2398554102055911763rm_a_b @ R9 @ top_to1314267278846557113rm_a_b ) ) ) ).
% SN_subset
thf(fact_625_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_626_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_627_SN__O__comm,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S4: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ R5 @ S4 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ R5 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_628_SN__O__comm,axiom,
! [R5: set_Pr6708357783813671845_b_nat,S4: set_Pr4549835640365387557rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom1324791570160179451rm_a_b @ R5 @ S4 ) @ top_top_set_term_a_b )
= ( abstract_SN_on_nat @ ( relcom152770009037377975_b_nat @ S4 @ R5 ) @ top_top_set_nat ) ) ).
% SN_O_comm
thf(fact_629_SN__O__comm,axiom,
! [R5: set_Pr4802440762858034464rm_a_b,S4: set_Pr6510905305361987970rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( relcom8422713090239460168rm_a_b @ R5 @ S4 ) @ top_top_set_term_a_b )
= ( abstra2398554102055911763rm_a_b @ ( relcom8086516547131752875rm_a_b @ S4 @ R5 ) @ top_to1314267278846557113rm_a_b ) ) ).
% SN_O_comm
thf(fact_630_SN__O__comm,axiom,
! [R5: set_Pr4549835640365387557rm_a_b,S4: set_Pr6708357783813671845_b_nat] :
( ( abstract_SN_on_nat @ ( relcom152770009037377975_b_nat @ R5 @ S4 ) @ top_top_set_nat )
= ( abstra4720023341729745482rm_a_b @ ( relcom1324791570160179451rm_a_b @ S4 @ R5 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_631_SN__O__comm,axiom,
! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ R5 @ S4 ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ S4 @ R5 ) @ top_top_set_nat ) ) ).
% SN_O_comm
thf(fact_632_SN__O__comm,axiom,
! [R5: set_Pr1899728212265713508rm_a_b,S4: set_Pr4414548462862917438_b_nat] :
( ( abstract_SN_on_nat @ ( relcom9080579859621841352_b_nat @ R5 @ S4 ) @ top_top_set_nat )
= ( abstra2398554102055911763rm_a_b @ ( relcom5491021006266299503rm_a_b @ S4 @ R5 ) @ top_to1314267278846557113rm_a_b ) ) ).
% SN_O_comm
thf(fact_633_SN__O__comm,axiom,
! [R5: set_Pr6510905305361987970rm_a_b,S4: set_Pr4802440762858034464rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ ( relcom8086516547131752875rm_a_b @ R5 @ S4 ) @ top_to1314267278846557113rm_a_b )
= ( abstra4720023341729745482rm_a_b @ ( relcom8422713090239460168rm_a_b @ S4 @ R5 ) @ top_top_set_term_a_b ) ) ).
% SN_O_comm
thf(fact_634_SN__O__comm,axiom,
! [R5: set_Pr4414548462862917438_b_nat,S4: set_Pr1899728212265713508rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ ( relcom5491021006266299503rm_a_b @ R5 @ S4 ) @ top_to1314267278846557113rm_a_b )
= ( abstract_SN_on_nat @ ( relcom9080579859621841352_b_nat @ S4 @ R5 ) @ top_top_set_nat ) ) ).
% SN_O_comm
thf(fact_635_SN__O__comm,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,S4: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ ( relcom6159724393391289352rm_a_b @ R5 @ S4 ) @ top_to1314267278846557113rm_a_b )
= ( abstra2398554102055911763rm_a_b @ ( relcom6159724393391289352rm_a_b @ S4 @ R5 ) @ top_to1314267278846557113rm_a_b ) ) ).
% SN_O_comm
thf(fact_636_SN__trancl__SN__conv,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R5 ) @ top_top_set_term_a_b )
= ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% SN_trancl_SN_conv
thf(fact_637_SN__trancl__SN__conv,axiom,
! [R5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R5 ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ R5 @ top_top_set_nat ) ) ).
% SN_trancl_SN_conv
thf(fact_638_SN__trancl__SN__conv,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ ( transi9211502839322181930rm_a_b @ R5 ) @ top_to1314267278846557113rm_a_b )
= ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b ) ) ).
% SN_trancl_SN_conv
thf(fact_639_SN__trancl__imp__SN,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R5 ) @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b ) ) ).
% SN_trancl_imp_SN
thf(fact_640_SN__trancl__imp__SN,axiom,
! [R5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R5 ) @ top_top_set_nat )
=> ( abstract_SN_on_nat @ R5 @ top_top_set_nat ) ) ).
% SN_trancl_imp_SN
thf(fact_641_SN__trancl__imp__SN,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ ( transi9211502839322181930rm_a_b @ R5 ) @ top_to1314267278846557113rm_a_b )
=> ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b ) ) ).
% SN_trancl_imp_SN
thf(fact_642_SN__imp__SN__trancl,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b )
=> ( abstra4720023341729745482rm_a_b @ ( transi7922773638565587891rm_a_b @ R5 ) @ top_top_set_term_a_b ) ) ).
% SN_imp_SN_trancl
thf(fact_643_SN__imp__SN__trancl,axiom,
! [R5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R5 @ top_top_set_nat )
=> ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R5 ) @ top_top_set_nat ) ) ).
% SN_imp_SN_trancl
thf(fact_644_SN__imp__SN__trancl,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b )
=> ( abstra2398554102055911763rm_a_b @ ( transi9211502839322181930rm_a_b @ R5 ) @ top_to1314267278846557113rm_a_b ) ) ).
% SN_imp_SN_trancl
thf(fact_645_SN__imp__WN,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( abstra4644735066934440262rm_a_b @ A @ top_top_set_term_a_b ) ) ).
% SN_imp_WN
thf(fact_646_SN__imp__WN,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( abstract_WN_on_nat @ A @ top_top_set_nat ) ) ).
% SN_imp_WN
thf(fact_647_SN__imp__WN,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( abstra2721873542170697047rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ) ).
% SN_imp_WN
thf(fact_648_SN__restrict__SN__idemp,axiom,
! [A: set_Pr4386577575007340137rm_a_b] : ( abstra4720023341729745482rm_a_b @ ( abstra4933154510623179783rm_a_b @ A @ A ) @ top_top_set_term_a_b ) ).
% SN_restrict_SN_idemp
thf(fact_649_SN__restrict__SN__idemp,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( abstract_SN_on_nat @ ( abstra2540617779794743107SN_nat @ A @ A ) @ top_top_set_nat ) ).
% SN_restrict_SN_idemp
thf(fact_650_SN__restrict__SN__idemp,axiom,
! [A: set_Pr2972776593051762503rm_a_b] : ( abstra2398554102055911763rm_a_b @ ( abstra5272529552662653398rm_a_b @ A @ A ) @ top_to1314267278846557113rm_a_b ) ).
% SN_restrict_SN_idemp
thf(fact_651_WCR__SN__on__imp__CR__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A: set_term_a_b] :
( ( abstra5026849107050860991rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra4720023341729745482rm_a_b @ R @ A )
=> ( abstra8448919418672941150rm_a_b @ R @ A ) ) ) ).
% WCR_SN_on_imp_CR_on
thf(fact_652_WCR__SN__on__imp__CR__on,axiom,
! [R: set_Pr1261947904930325089at_nat,A: set_nat] :
( ( abstract_WCR_on_nat @ R @ top_top_set_nat )
=> ( ( abstract_SN_on_nat @ R @ A )
=> ( abstract_CR_on_nat @ R @ A ) ) ) ).
% WCR_SN_on_imp_CR_on
thf(fact_653_WCR__SN__on__imp__CR__on,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( abstra875157153669124638rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( abstra2398554102055911763rm_a_b @ R @ A )
=> ( abstra7359694574113289279rm_a_b @ R @ A ) ) ) ).
% WCR_SN_on_imp_CR_on
thf(fact_654_Newman,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra5026849107050860991rm_a_b @ R @ top_top_set_term_a_b )
=> ( abstra8448919418672941150rm_a_b @ R @ top_top_set_term_a_b ) ) ) ).
% Newman
thf(fact_655_Newman,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( ( abstract_WCR_on_nat @ R @ top_top_set_nat )
=> ( abstract_CR_on_nat @ R @ top_top_set_nat ) ) ) ).
% Newman
thf(fact_656_Newman,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( abstra875157153669124638rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( abstra7359694574113289279rm_a_b @ R @ top_to1314267278846557113rm_a_b ) ) ) ).
% Newman
thf(fact_657_quasi__commute__imp__SN,axiom,
! [R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( ( abstra4720023341729745482rm_a_b @ S @ top_top_set_term_a_b )
=> ( ( abstra4317384144805385751rm_a_b @ R @ S )
=> ( abstra4720023341729745482rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ S ) @ top_top_set_term_a_b ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_658_quasi__commute__imp__SN,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( ( abstract_SN_on_nat @ S @ top_top_set_nat )
=> ( ( abstra4838182543893801299te_nat @ R @ S )
=> ( abstract_SN_on_nat @ ( sup_su6327502436637775413at_nat @ R @ S ) @ top_top_set_nat ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_659_quasi__commute__imp__SN,axiom,
! [R: set_Pr2972776593051762503rm_a_b,S: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( ( abstra2398554102055911763rm_a_b @ S @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7274158693546625990rm_a_b @ R @ S )
=> ( abstra2398554102055911763rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R @ S ) @ top_to1314267278846557113rm_a_b ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_660_compatible__SN,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( abstra4720023341729745482rm_a_b @ ( relcom370159955682700863rm_a_b @ S4 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S4 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ top_top_set_term_a_b ) ) ) ).
% compatible_SN
thf(fact_661_compatible__SN,axiom,
! [S4: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ S4 @ top_top_set_nat )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S4 ) @ S4 )
=> ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ S4 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S4 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) @ top_top_set_nat ) ) ) ).
% compatible_SN
thf(fact_662_compatible__SN,axiom,
! [S4: set_Pr2972776593051762503rm_a_b,NS: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ S4 @ top_to1314267278846557113rm_a_b )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ NS @ S4 ) @ S4 )
=> ( abstra2398554102055911763rm_a_b @ ( relcom6159724393391289352rm_a_b @ S4 @ ( relcom6159724393391289352rm_a_b @ ( transi2615809358984392588rm_a_b @ S4 ) @ ( transi2615809358984392588rm_a_b @ NS ) ) ) @ top_to1314267278846557113rm_a_b ) ) ) ).
% compatible_SN
thf(fact_663_ideriv__SN,axiom,
! [S4: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,As: nat > term_a_b] :
( ( abstra4720023341729745482rm_a_b @ S4 @ top_top_set_term_a_b )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( ord_le118470702582115849rm_a_b @ R5 @ ( sup_su6776935440552674877rm_a_b @ NS @ S4 ) )
=> ~ ( abstra5386884192082944997rm_a_b @ ( inf_in1693821330460682071rm_a_b @ S4 @ R5 ) @ ( minus_5192120951422937424rm_a_b @ R5 @ S4 ) @ As ) ) ) ) ).
% ideriv_SN
thf(fact_664_ideriv__SN,axiom,
! [S4: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,As: nat > nat] :
( ( abstract_SN_on_nat @ S4 @ top_top_set_nat )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S4 ) @ S4 )
=> ( ( ord_le3146513528884898305at_nat @ R5 @ ( sup_su6327502436637775413at_nat @ NS @ S4 ) )
=> ~ ( abstract_ideriv_nat @ ( inf_in2572325071724192079at_nat @ S4 @ R5 ) @ ( minus_1356011639430497352at_nat @ R5 @ S4 ) @ As ) ) ) ) ).
% ideriv_SN
thf(fact_665_ideriv__SN,axiom,
! [S4: set_Pr2972776593051762503rm_a_b,NS: set_Pr2972776593051762503rm_a_b,R5: set_Pr2972776593051762503rm_a_b,As: nat > produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ S4 @ top_to1314267278846557113rm_a_b )
=> ( ( ord_le2593958535671617255rm_a_b @ ( relcom6159724393391289352rm_a_b @ NS @ S4 ) @ S4 )
=> ( ( ord_le2593958535671617255rm_a_b @ R5 @ ( sup_su1845815533690797339rm_a_b @ NS @ S4 ) )
=> ~ ( abstra4518033177758647160rm_a_b @ ( inf_in7724013137991509301rm_a_b @ S4 @ R5 ) @ ( minus_45395518312058030rm_a_b @ R5 @ S4 ) @ As ) ) ) ) ).
% ideriv_SN
thf(fact_666_the__NF,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( abstra2242782261322123075rm_a_b @ A @ A2 ) ) @ ( abstra2972651231392952684rm_a_b @ A ) ) ) ) ).
% the_NF
thf(fact_667_the__NF,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( ( abstract_CR_on_nat @ A @ top_top_set_nat )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( abstract_the_NF_nat @ A @ A2 ) ) @ ( abstra7312431243108365096ty_nat @ A ) ) ) ) ).
% the_NF
thf(fact_668_the__NF,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( abstra8972549633426078106rm_a_b @ A @ A2 ) ) @ ( abstra1496011390211842225rm_a_b @ A ) ) ) ) ).
% the_NF
thf(fact_669_the__NF__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( abstra1130568886744591910rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_670_the__NF__conv,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( ( abstract_CR_on_nat @ A @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( abstra2375765306220057826on_nat @ A ) )
=> ( ( abstract_the_NF_nat @ A @ A2 )
= ( abstract_the_NF_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_671_the__NF__conv,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( abstra6125656530262095991rm_a_b @ A ) )
=> ( ( abstra8972549633426078106rm_a_b @ A @ A2 )
= ( abstra8972549633426078106rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_conv
thf(fact_672_the__NF__step,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ A )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_673_the__NF__step,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( ( abstract_CR_on_nat @ A @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ A )
=> ( ( abstract_the_NF_nat @ A @ A2 )
= ( abstract_the_NF_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_674_the__NF__step,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ A )
=> ( ( abstra8972549633426078106rm_a_b @ A @ A2 )
= ( abstra8972549633426078106rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_step
thf(fact_675_the__NF__NF,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( member_term_a_b @ ( abstra2242782261322123075rm_a_b @ A @ A2 ) @ ( abstract_NF_term_a_b @ A ) ) ) ) ).
% the_NF_NF
thf(fact_676_the__NF__NF,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( ( abstract_CR_on_nat @ A @ top_top_set_nat )
=> ( member_nat @ ( abstract_the_NF_nat @ A @ A2 ) @ ( abstract_NF_nat @ A ) ) ) ) ).
% the_NF_NF
thf(fact_677_the__NF__NF,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( member5869715511025134514rm_a_b @ ( abstra8972549633426078106rm_a_b @ A @ A2 ) @ ( abstra1902783451776770197rm_a_b @ A ) ) ) ) ).
% the_NF_NF
thf(fact_678_the__NF__steps,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( abstra8448919418672941150rm_a_b @ A @ top_top_set_term_a_b )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra2242782261322123075rm_a_b @ A @ A2 )
= ( abstra2242782261322123075rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_679_the__NF__steps,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
=> ( ( abstract_CR_on_nat @ A @ top_top_set_nat )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ A ) )
=> ( ( abstract_the_NF_nat @ A @ A2 )
= ( abstract_the_NF_nat @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_680_the__NF__steps,axiom,
! [A: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( abstra7359694574113289279rm_a_b @ A @ top_to1314267278846557113rm_a_b )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ ( transi2615809358984392588rm_a_b @ A ) )
=> ( ( abstra8972549633426078106rm_a_b @ A @ A2 )
= ( abstra8972549633426078106rm_a_b @ A @ B2 ) ) ) ) ) ).
% the_NF_steps
thf(fact_681_irrefl__tranclI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( conver7608559976937845093rm_a_b @ R ) @ ( transi7742714808557438673rm_a_b @ R ) )
= bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ ( transi7922773638565587891rm_a_b @ R ) ) ) ).
% irrefl_tranclI
thf(fact_682_irrefl__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( irrefl_on_term_a_b @ top_top_set_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ).
% irrefl_diff_Id
thf(fact_683_irrefl__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( irrefl_on_nat @ top_top_set_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ).
% irrefl_diff_Id
thf(fact_684_irrefl__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b] : ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ).
% irrefl_diff_Id
thf(fact_685_empty__iff,axiom,
! [C: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_686_empty__iff,axiom,
! [C: term_a_b] :
~ ( member_term_a_b @ C @ bot_bot_set_term_a_b ) ).
% empty_iff
thf(fact_687_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_688_all__not__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ! [X4: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_689_all__not__in__conv,axiom,
! [A: set_term_a_b] :
( ( ! [X4: term_a_b] :
~ ( member_term_a_b @ X4 @ A ) )
= ( A = bot_bot_set_term_a_b ) ) ).
% all_not_in_conv
thf(fact_690_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_691_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_692_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X4: nat] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_693_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_694_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_695_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_696_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_697_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_698_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_699_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_700_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_701_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_702_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_703_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_704_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_705_Un__empty,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( sup_sup_set_nat @ A @ B6 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B6 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_706_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_707_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_708_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_709_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_710_Diff__UNIV,axiom,
! [A: set_term_a_b] :
( ( minus_4613853266328121507rm_a_b @ A @ top_top_set_term_a_b )
= bot_bot_set_term_a_b ) ).
% Diff_UNIV
thf(fact_711_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_712_Diff__UNIV,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( minus_5192120951422937424rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% Diff_UNIV
thf(fact_713_Diff__eq__empty__iff,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( minus_minus_set_nat @ A @ B6 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_714_Diff__disjoint,axiom,
! [A: set_nat,B6: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B6 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_715_NF__empty,axiom,
( ( abstract_NF_term_a_b @ bot_bo197521221353338581rm_a_b )
= top_top_set_term_a_b ) ).
% NF_empty
thf(fact_716_NF__empty,axiom,
( ( abstract_NF_nat @ bot_bo2099793752762293965at_nat )
= top_top_set_nat ) ).
% NF_empty
thf(fact_717_NF__empty,axiom,
( ( abstra1902783451776770197rm_a_b @ bot_bo5040873672931234227rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% NF_empty
thf(fact_718_Int__emptyI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X2 @ B6 ) )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ B6 )
= bot_bo197521221353338581rm_a_b ) ) ).
% Int_emptyI
thf(fact_719_Int__emptyI,axiom,
! [A: set_term_a_b,B6: set_term_a_b] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ~ ( member_term_a_b @ X2 @ B6 ) )
=> ( ( inf_inf_set_term_a_b @ A @ B6 )
= bot_bot_set_term_a_b ) ) ).
% Int_emptyI
thf(fact_720_Int__emptyI,axiom,
! [A: set_nat,B6: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B6 ) )
=> ( ( inf_inf_set_nat @ A @ B6 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_721_disjoint__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A @ B6 )
= bot_bo197521221353338581rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ A )
=> ~ ( member5869715511025134514rm_a_b @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_722_disjoint__iff,axiom,
! [A: set_term_a_b,B6: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A @ B6 )
= bot_bot_set_term_a_b )
= ( ! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A )
=> ~ ( member_term_a_b @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_723_disjoint__iff,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( inf_inf_set_nat @ A @ B6 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_724_Int__empty__left,axiom,
! [B6: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B6 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_725_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_726_disjoint__iff__not__equal,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( inf_inf_set_nat @ A @ B6 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ B6 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_727_irrefl__onD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_728_irrefl__onD,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( irrefl_on_nat @ A @ R )
=> ( ( member_nat @ A2 @ A )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_729_irrefl__onD,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( irrefl_on_term_a_b @ A @ R )
=> ( ( member_term_a_b @ A2 @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% irrefl_onD
thf(fact_730_irrefl__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A4 @ A )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R ) )
=> ( irrefl6620156600162922239rm_a_b @ A @ R ) ) ).
% irrefl_onI
thf(fact_731_irrefl__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_nat @ A @ R ) ) ).
% irrefl_onI
thf(fact_732_irrefl__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
( ( member_term_a_b @ A4 @ A )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R ) )
=> ( irrefl_on_term_a_b @ A @ R ) ) ).
% irrefl_onI
thf(fact_733_irrefl__on__def,axiom,
( irrefl_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ R4 ) ) ) ) ).
% irrefl_on_def
thf(fact_734_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_735_emptyE,axiom,
! [A2: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_736_emptyE,axiom,
! [A2: term_a_b] :
~ ( member_term_a_b @ A2 @ bot_bot_set_term_a_b ) ).
% emptyE
thf(fact_737_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_738_equals0D,axiom,
! [A: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( A = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_739_equals0D,axiom,
! [A: set_term_a_b,A2: term_a_b] :
( ( A = bot_bot_set_term_a_b )
=> ~ ( member_term_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_740_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_741_equals0I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y3 @ A )
=> ( A = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_742_equals0I,axiom,
! [A: set_term_a_b] :
( ! [Y3: term_a_b] :
~ ( member_term_a_b @ Y3 @ A )
=> ( A = bot_bot_set_term_a_b ) ) ).
% equals0I
thf(fact_743_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_744_ex__in__conv,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ A ) )
= ( A != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_745_ex__in__conv,axiom,
! [A: set_term_a_b] :
( ( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ A ) )
= ( A != bot_bot_set_term_a_b ) ) ).
% ex_in_conv
thf(fact_746_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_747_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_748_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( X4 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_749_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_750_Un__empty__left,axiom,
! [B6: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B6 )
= B6 ) ).
% Un_empty_left
thf(fact_751_empty__not__UNIV,axiom,
bot_bot_set_term_a_b != top_top_set_term_a_b ).
% empty_not_UNIV
thf(fact_752_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_753_empty__not__UNIV,axiom,
bot_bo197521221353338581rm_a_b != top_to1314267278846557113rm_a_b ).
% empty_not_UNIV
thf(fact_754_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_755_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_756_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_757_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_758_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_759_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_760_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_761_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_762_cSup__eq__non__empty,axiom,
! [X3: set_nat,A2: nat] :
( ( X3 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X3 )
=> ( ord_less_eq_nat @ X2 @ A2 ) )
=> ( ! [Y3: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X3 )
=> ( ord_less_eq_nat @ X6 @ Y3 ) )
=> ( ord_less_eq_nat @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X3 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_763_cSup__least,axiom,
! [X3: set_nat,Z2: nat] :
( ( X3 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X3 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X3 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_764_Union__disjoint,axiom,
! [C2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ C2 )
=> ( ( inf_inf_set_nat @ X4 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_765_Int__Diff__disjoint,axiom,
! [A: set_nat,B6: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B6 ) @ ( minus_minus_set_nat @ A @ B6 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_766_Diff__triv,axiom,
! [A: set_nat,B6: set_nat] :
( ( ( inf_inf_set_nat @ A @ B6 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B6 )
= A ) ) ).
% Diff_triv
thf(fact_767_SN__empty,axiom,
abstra4720023341729745482rm_a_b @ bot_bo197521221353338581rm_a_b @ top_top_set_term_a_b ).
% SN_empty
thf(fact_768_SN__empty,axiom,
abstract_SN_on_nat @ bot_bo2099793752762293965at_nat @ top_top_set_nat ).
% SN_empty
thf(fact_769_SN__empty,axiom,
abstra2398554102055911763rm_a_b @ bot_bo5040873672931234227rm_a_b @ top_to1314267278846557113rm_a_b ).
% SN_empty
thf(fact_770_irreflI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ R )
=> ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% irreflI
thf(fact_771_irreflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
=> ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% irreflI
thf(fact_772_irreflI,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] :
~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A4 @ A4 ) @ R )
=> ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% irreflI
thf(fact_773_irreflD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R ) ) ).
% irreflD
thf(fact_774_irreflD,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( irrefl_on_nat @ top_top_set_nat @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) ) ).
% irreflD
thf(fact_775_irreflD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ X ) @ R ) ) ).
% irreflD
thf(fact_776_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_term_a_b,X: set_term_a_b,Y: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ A2 @ X )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A2 @ X )
= top_top_set_term_a_b )
=> ( ( ( inf_inf_set_term_a_b @ A2 @ Y )
= bot_bot_set_term_a_b )
=> ( ( ( sup_sup_set_term_a_b @ A2 @ Y )
= top_top_set_term_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_777_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_778_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Pr4386577575007340137rm_a_b,X: set_Pr4386577575007340137rm_a_b,Y: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ A2 @ X )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A2 @ X )
= top_to1314267278846557113rm_a_b )
=> ( ( ( inf_in1693821330460682071rm_a_b @ A2 @ Y )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( sup_su6776935440552674877rm_a_b @ A2 @ Y )
= top_to1314267278846557113rm_a_b )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_779_Sup__inf__eq__bot__iff,axiom,
! [B6: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B6 ) @ A2 )
= bot_bot_set_nat )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B6 )
=> ( ( inf_inf_set_nat @ X4 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_780_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_781_bot__empty__eq,axiom,
( bot_bot_term_a_b_o
= ( ^ [X4: term_a_b] : ( member_term_a_b @ X4 @ bot_bot_set_term_a_b ) ) ) ).
% bot_empty_eq
thf(fact_782_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_783_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_784_some__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ ( abstra8203652949708803681rm_a_b @ R @ X ) ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ ( abstra8203652949708803681rm_a_b @ R @ X ) @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_785_some__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ ( abstra2589370493620291516rm_a_b @ R @ X ) ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ ( abstra2589370493620291516rm_a_b @ R @ X ) @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% some_NF
thf(fact_786_some__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ ( abstract_some_NF_nat @ R @ X ) ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ ( abstract_some_NF_nat @ R @ X ) @ ( abstract_NF_nat @ R ) ) ) ) ).
% some_NF
thf(fact_787_CR__on__singletonI,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b,C4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ C4 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C4 ) @ ( abstra4096080454567261402rm_a_b @ R ) ) ) )
=> ( abstra8448919418672941150rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% CR_on_singletonI
thf(fact_788_CR__on__singletonI,axiom,
! [A2: nat,R: set_Pr1261947904930325089at_nat] :
( ! [B: nat,C4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C4 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C4 ) @ ( abstract_join_nat @ R ) ) ) )
=> ( abstract_CR_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% CR_on_singletonI
thf(fact_789_SN__reaches__NF,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ? [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y3 ) @ ( transi2615809358984392588rm_a_b @ R ) )
& ( member5869715511025134514rm_a_b @ Y3 @ ( abstra1902783451776770197rm_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_790_SN__reaches__NF,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ? [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R ) )
& ( member_term_a_b @ Y3 @ ( abstract_NF_term_a_b @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_791_SN__reaches__NF,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ? [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member_nat @ Y3 @ ( abstract_NF_nat @ R ) ) ) ) ).
% SN_reaches_NF
thf(fact_792_insertCI,axiom,
! [A2: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ A2 @ B6 )
=> ( A2 = B2 ) )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B6 ) ) ) ).
% insertCI
thf(fact_793_insertCI,axiom,
! [A2: nat,B6: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B6 )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B6 ) ) ) ).
% insertCI
thf(fact_794_insertCI,axiom,
! [A2: term_a_b,B6: set_term_a_b,B2: term_a_b] :
( ( ~ ( member_term_a_b @ A2 @ B6 )
=> ( A2 = B2 ) )
=> ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ B6 ) ) ) ).
% insertCI
thf(fact_795_insert__iff,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_796_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_797_insert__iff,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_term_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_798_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_799_singletonI,axiom,
! [A2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ).
% singletonI
thf(fact_800_singletonI,axiom,
! [A2: term_a_b] : ( member_term_a_b @ A2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ).
% singletonI
thf(fact_801_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_802_insert__subset,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A ) @ B6 )
= ( ( member5869715511025134514rm_a_b @ X @ B6 )
& ( ord_le118470702582115849rm_a_b @ A @ B6 ) ) ) ).
% insert_subset
thf(fact_803_insert__subset,axiom,
! [X: nat,A: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B6 )
= ( ( member_nat @ X @ B6 )
& ( ord_less_eq_set_nat @ A @ B6 ) ) ) ).
% insert_subset
thf(fact_804_insert__subset,axiom,
! [X: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ X @ A ) @ B6 )
= ( ( member_term_a_b @ X @ B6 )
& ( ord_le2705286416250468010rm_a_b @ A @ B6 ) ) ) ).
% insert_subset
thf(fact_805_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B6: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B6 ) ) ) ).
% Un_insert_right
thf(fact_806_Un__insert__left,axiom,
! [A2: nat,B6: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B6 ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B6 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_807_Int__insert__left__if0,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B6 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_808_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B6: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B6 ) @ C2 )
= ( inf_inf_set_nat @ B6 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_809_Int__insert__left__if0,axiom,
! [A2: term_a_b,C2: set_term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B6 ) @ C2 )
= ( inf_inf_set_term_a_b @ B6 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_810_Int__insert__left__if1,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B6 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_811_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B6: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B6 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B6 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_812_Int__insert__left__if1,axiom,
! [A2: term_a_b,C2: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B6 ) @ C2 )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ B6 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_813_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B6 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B6 ) ) ) ).
% insert_inter_insert
thf(fact_814_Int__insert__right__if0,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_815_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( inf_inf_set_nat @ A @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_816_Int__insert__right__if0,axiom,
! [A2: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B6 ) )
= ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_817_Int__insert__right__if1,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_818_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_819_Int__insert__right__if1,axiom,
! [A2: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B6 ) )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_820_insert__Diff1,axiom,
! [X: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ B6 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A ) @ B6 )
= ( minus_5192120951422937424rm_a_b @ A @ B6 ) ) ) ).
% insert_Diff1
thf(fact_821_insert__Diff1,axiom,
! [X: nat,B6: set_nat,A: set_nat] :
( ( member_nat @ X @ B6 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B6 )
= ( minus_minus_set_nat @ A @ B6 ) ) ) ).
% insert_Diff1
thf(fact_822_insert__Diff1,axiom,
! [X: term_a_b,B6: set_term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ X @ B6 )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B6 )
= ( minus_4613853266328121507rm_a_b @ A @ B6 ) ) ) ).
% insert_Diff1
thf(fact_823_Diff__insert0,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X @ B6 ) )
= ( minus_5192120951422937424rm_a_b @ A @ B6 ) ) ) ).
% Diff_insert0
thf(fact_824_Diff__insert0,axiom,
! [X: nat,A: set_nat,B6: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B6 ) )
= ( minus_minus_set_nat @ A @ B6 ) ) ) ).
% Diff_insert0
thf(fact_825_Diff__insert0,axiom,
! [X: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ X @ B6 ) )
= ( minus_4613853266328121507rm_a_b @ A @ B6 ) ) ) ).
% Diff_insert0
thf(fact_826_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_827_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_828_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_829_disjoint__insert_I2_J,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ B2 @ B6 ) ) )
= ( ~ ( member5869715511025134514rm_a_b @ B2 @ A )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_830_disjoint__insert_I2_J,axiom,
! [A: set_term_a_b,B2: term_a_b,B6: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ B2 @ B6 ) ) )
= ( ~ ( member_term_a_b @ B2 @ A )
& ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_831_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B6: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B6 ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_832_disjoint__insert_I1_J,axiom,
! [B6: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ B6 @ ( insert7009541432154983385rm_a_b @ A2 @ A ) )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B6 )
& ( ( inf_in1693821330460682071rm_a_b @ B6 @ A )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_833_disjoint__insert_I1_J,axiom,
! [B6: set_term_a_b,A2: term_a_b,A: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ B6 @ ( insert_term_a_b @ A2 @ A ) )
= bot_bot_set_term_a_b )
= ( ~ ( member_term_a_b @ A2 @ B6 )
& ( ( inf_inf_set_term_a_b @ B6 @ A )
= bot_bot_set_term_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_834_disjoint__insert_I1_J,axiom,
! [B6: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B6 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B6 )
& ( ( inf_inf_set_nat @ B6 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_835_insert__disjoint_I2_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B6 ) )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B6 )
& ( bot_bo197521221353338581rm_a_b
= ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_836_insert__disjoint_I2_J,axiom,
! [A2: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ A ) @ B6 ) )
= ( ~ ( member_term_a_b @ A2 @ B6 )
& ( bot_bot_set_term_a_b
= ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_837_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B6 ) )
= ( ~ ( member_nat @ A2 @ B6 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_838_insert__disjoint_I1_J,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ A ) @ B6 )
= bot_bo197521221353338581rm_a_b )
= ( ~ ( member5869715511025134514rm_a_b @ A2 @ B6 )
& ( ( inf_in1693821330460682071rm_a_b @ A @ B6 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_839_insert__disjoint_I1_J,axiom,
! [A2: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ A ) @ B6 )
= bot_bot_set_term_a_b )
= ( ~ ( member_term_a_b @ A2 @ B6 )
& ( ( inf_inf_set_term_a_b @ A @ B6 )
= bot_bot_set_term_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_840_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B6 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B6 )
& ( ( inf_inf_set_nat @ A @ B6 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_841_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_842_singletonD,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_843_singletonD,axiom,
! [B2: term_a_b,A2: term_a_b] :
( ( member_term_a_b @ B2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_844_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_845_singleton__iff,axiom,
! [B2: produc357393685978478089rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B2 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_846_singleton__iff,axiom,
! [B2: term_a_b,A2: term_a_b] :
( ( member_term_a_b @ B2 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_847_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_848_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_849_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_850_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_851_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_852_subset__singleton__iff,axiom,
! [X3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
| ( X3
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_853_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_854_Un__singleton__iff,axiom,
! [A: set_nat,B6: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B6 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B6
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B6 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B6
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_855_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B6: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B6 ) )
= ( ( ( A = bot_bot_set_nat )
& ( B6
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B6 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B6
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_856_Diff__insert__absorb,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A ) @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_857_Diff__insert__absorb,axiom,
! [X: term_a_b,A: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_858_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_859_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B6: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_860_insert__Diff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_861_insert__Diff,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( insert_term_a_b @ A2 @ ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_862_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_863_Diff__insert,axiom,
! [A: set_nat,A2: nat,B6: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B6 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_864_subset__Diff__insert,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B6 @ ( insert7009541432154983385rm_a_b @ X @ C2 ) ) )
= ( ( ord_le118470702582115849rm_a_b @ A @ ( minus_5192120951422937424rm_a_b @ B6 @ C2 ) )
& ~ ( member5869715511025134514rm_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_865_subset__Diff__insert,axiom,
! [A: set_nat,B6: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B6 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B6 @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_866_subset__Diff__insert,axiom,
! [A: set_term_a_b,B6: set_term_a_b,X: term_a_b,C2: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ ( minus_4613853266328121507rm_a_b @ B6 @ ( insert_term_a_b @ X @ C2 ) ) )
= ( ( ord_le2705286416250468010rm_a_b @ A @ ( minus_4613853266328121507rm_a_b @ B6 @ C2 ) )
& ~ ( member_term_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_867_Int__insert__left,axiom,
! [A2: produc357393685978478089rm_a_b,C2: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) @ C2 )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ B6 @ C2 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ C2 )
=> ( ( inf_in1693821330460682071rm_a_b @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) @ C2 )
= ( inf_in1693821330460682071rm_a_b @ B6 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_868_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B6: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B6 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B6 @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B6 ) @ C2 )
= ( inf_inf_set_nat @ B6 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_869_Int__insert__left,axiom,
! [A2: term_a_b,C2: set_term_a_b,B6: set_term_a_b] :
( ( ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B6 ) @ C2 )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ B6 @ C2 ) ) ) )
& ( ~ ( member_term_a_b @ A2 @ C2 )
=> ( ( inf_inf_set_term_a_b @ ( insert_term_a_b @ A2 @ B6 ) @ C2 )
= ( inf_inf_set_term_a_b @ B6 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_870_Int__insert__right,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) )
= ( insert7009541432154983385rm_a_b @ A2 @ ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( inf_in1693821330460682071rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) )
= ( inf_in1693821330460682071rm_a_b @ A @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_871_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B6: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B6 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B6 ) )
= ( inf_inf_set_nat @ A @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_872_Int__insert__right,axiom,
! [A2: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ( ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B6 ) )
= ( insert_term_a_b @ A2 @ ( inf_inf_set_term_a_b @ A @ B6 ) ) ) )
& ( ~ ( member_term_a_b @ A2 @ A )
=> ( ( inf_inf_set_term_a_b @ A @ ( insert_term_a_b @ A2 @ B6 ) )
= ( inf_inf_set_term_a_b @ A @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_873_insertE,axiom,
! [A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member5869715511025134514rm_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_874_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_875_insertE,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_term_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_876_insertI1,axiom,
! [A2: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ A2 @ B6 ) ) ).
% insertI1
thf(fact_877_insertI1,axiom,
! [A2: nat,B6: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B6 ) ) ).
% insertI1
thf(fact_878_insertI1,axiom,
! [A2: term_a_b,B6: set_term_a_b] : ( member_term_a_b @ A2 @ ( insert_term_a_b @ A2 @ B6 ) ) ).
% insertI1
thf(fact_879_insertI2,axiom,
! [A2: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ B6 )
=> ( member5869715511025134514rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ B6 ) ) ) ).
% insertI2
thf(fact_880_insertI2,axiom,
! [A2: nat,B6: set_nat,B2: nat] :
( ( member_nat @ A2 @ B6 )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B6 ) ) ) ).
% insertI2
thf(fact_881_insertI2,axiom,
! [A2: term_a_b,B6: set_term_a_b,B2: term_a_b] :
( ( member_term_a_b @ A2 @ B6 )
=> ( member_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ B6 ) ) ) ).
% insertI2
thf(fact_882_Set_Oset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ~ ! [B8: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ X @ B8 ) )
=> ( member5869715511025134514rm_a_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_883_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B8: set_nat] :
( ( A
= ( insert_nat @ X @ B8 ) )
=> ( member_nat @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_884_Set_Oset__insert,axiom,
! [X: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ X @ A )
=> ~ ! [B8: set_term_a_b] :
( ( A
= ( insert_term_a_b @ X @ B8 ) )
=> ( member_term_a_b @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_885_insert__ident,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ X @ B6 )
=> ( ( ( insert7009541432154983385rm_a_b @ X @ A )
= ( insert7009541432154983385rm_a_b @ X @ B6 ) )
= ( A = B6 ) ) ) ) ).
% insert_ident
thf(fact_886_insert__ident,axiom,
! [X: nat,A: set_nat,B6: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B6 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B6 ) )
= ( A = B6 ) ) ) ) ).
% insert_ident
thf(fact_887_insert__ident,axiom,
! [X: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ~ ( member_term_a_b @ X @ B6 )
=> ( ( ( insert_term_a_b @ X @ A )
= ( insert_term_a_b @ X @ B6 ) )
= ( A = B6 ) ) ) ) ).
% insert_ident
thf(fact_888_insert__absorb,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_889_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_890_insert__absorb,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ( ( insert_term_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_891_insert__eq__iff,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( ~ ( member5869715511025134514rm_a_b @ B2 @ B6 )
=> ( ( ( insert7009541432154983385rm_a_b @ A2 @ A )
= ( insert7009541432154983385rm_a_b @ B2 @ B6 ) )
= ( ( ( A2 = B2 )
=> ( A = B6 ) )
& ( ( A2 != B2 )
=> ? [C5: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ B2 @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ B2 @ C5 )
& ( B6
= ( insert7009541432154983385rm_a_b @ A2 @ C5 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_892_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B6: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B6 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B6 ) )
= ( ( ( A2 = B2 )
=> ( A = B6 ) )
& ( ( A2 != B2 )
=> ? [C5: set_nat] :
( ( A
= ( insert_nat @ B2 @ C5 ) )
& ~ ( member_nat @ B2 @ C5 )
& ( B6
= ( insert_nat @ A2 @ C5 ) )
& ~ ( member_nat @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_893_insert__eq__iff,axiom,
! [A2: term_a_b,A: set_term_a_b,B2: term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ A2 @ A )
=> ( ~ ( member_term_a_b @ B2 @ B6 )
=> ( ( ( insert_term_a_b @ A2 @ A )
= ( insert_term_a_b @ B2 @ B6 ) )
= ( ( ( A2 = B2 )
=> ( A = B6 ) )
& ( ( A2 != B2 )
=> ? [C5: set_term_a_b] :
( ( A
= ( insert_term_a_b @ B2 @ C5 ) )
& ~ ( member_term_a_b @ B2 @ C5 )
& ( B6
= ( insert_term_a_b @ A2 @ C5 ) )
& ~ ( member_term_a_b @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_894_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_895_mk__disjoint__insert,axiom,
! [A2: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ? [B8: set_Pr4386577575007340137rm_a_b] :
( ( A
= ( insert7009541432154983385rm_a_b @ A2 @ B8 ) )
& ~ ( member5869715511025134514rm_a_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_896_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B8: set_nat] :
( ( A
= ( insert_nat @ A2 @ B8 ) )
& ~ ( member_nat @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_897_mk__disjoint__insert,axiom,
! [A2: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ A2 @ A )
=> ? [B8: set_term_a_b] :
( ( A
= ( insert_term_a_b @ A2 @ B8 ) )
& ~ ( member_term_a_b @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_898_insert__UNIV,axiom,
! [X: term_a_b] :
( ( insert_term_a_b @ X @ top_top_set_term_a_b )
= top_top_set_term_a_b ) ).
% insert_UNIV
thf(fact_899_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_900_insert__UNIV,axiom,
! [X: produc357393685978478089rm_a_b] :
( ( insert7009541432154983385rm_a_b @ X @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b ) ).
% insert_UNIV
thf(fact_901_insert__mono,axiom,
! [C2: set_nat,D3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_902_subset__insert,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X @ B6 ) )
= ( ord_le118470702582115849rm_a_b @ A @ B6 ) ) ) ).
% subset_insert
thf(fact_903_subset__insert,axiom,
! [X: nat,A: set_nat,B6: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B6 ) )
= ( ord_less_eq_set_nat @ A @ B6 ) ) ) ).
% subset_insert
thf(fact_904_subset__insert,axiom,
! [X: term_a_b,A: set_term_a_b,B6: set_term_a_b] :
( ~ ( member_term_a_b @ X @ A )
=> ( ( ord_le2705286416250468010rm_a_b @ A @ ( insert_term_a_b @ X @ B6 ) )
= ( ord_le2705286416250468010rm_a_b @ A @ B6 ) ) ) ).
% subset_insert
thf(fact_905_subset__insertI,axiom,
! [B6: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B6 @ ( insert_nat @ A2 @ B6 ) ) ).
% subset_insertI
thf(fact_906_subset__insertI2,axiom,
! [A: set_nat,B6: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B6 ) ) ) ).
% subset_insertI2
thf(fact_907_insert__Diff__if,axiom,
! [X: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X @ B6 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A ) @ B6 )
= ( minus_5192120951422937424rm_a_b @ A @ B6 ) ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ B6 )
=> ( ( minus_5192120951422937424rm_a_b @ ( insert7009541432154983385rm_a_b @ X @ A ) @ B6 )
= ( insert7009541432154983385rm_a_b @ X @ ( minus_5192120951422937424rm_a_b @ A @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_908_insert__Diff__if,axiom,
! [X: nat,B6: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B6 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B6 )
= ( minus_minus_set_nat @ A @ B6 ) ) )
& ( ~ ( member_nat @ X @ B6 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B6 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_909_insert__Diff__if,axiom,
! [X: term_a_b,B6: set_term_a_b,A: set_term_a_b] :
( ( ( member_term_a_b @ X @ B6 )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B6 )
= ( minus_4613853266328121507rm_a_b @ A @ B6 ) ) )
& ( ~ ( member_term_a_b @ X @ B6 )
=> ( ( minus_4613853266328121507rm_a_b @ ( insert_term_a_b @ X @ A ) @ B6 )
= ( insert_term_a_b @ X @ ( minus_4613853266328121507rm_a_b @ A @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_910_subset__insert__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X @ B6 ) )
= ( ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ord_le118470702582115849rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B6 ) )
& ( ~ ( member5869715511025134514rm_a_b @ X @ A )
=> ( ord_le118470702582115849rm_a_b @ A @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_911_subset__insert__iff,axiom,
! [A: set_term_a_b,X: term_a_b,B6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A @ ( insert_term_a_b @ X @ B6 ) )
= ( ( ( member_term_a_b @ X @ A )
=> ( ord_le2705286416250468010rm_a_b @ ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) ) @ B6 ) )
& ( ~ ( member_term_a_b @ X @ A )
=> ( ord_le2705286416250468010rm_a_b @ A @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_912_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B6 ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B6 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_913_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B6 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_914_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
= ( ! [B4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B4 ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B4 @ bot_bot_set_term_a_b ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_915_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_916_lower__set__imp__not__SN__on,axiom,
! [S: produc357393685978478089rm_a_b,X3: set_Pr4386577575007340137rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ S @ X3 )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ X3 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ X3 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ S @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_917_lower__set__imp__not__SN__on,axiom,
! [S: term_a_b,X3: set_term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ S @ X3 )
=> ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ X3 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ X3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_918_lower__set__imp__not__SN__on,axiom,
! [S: nat,X3: set_nat,R5: set_Pr1261947904930325089at_nat] :
( ( member_nat @ S @ X3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Xa2 ) @ R5 ) ) )
=> ~ ( abstract_SN_on_nat @ R5 @ ( insert_nat @ S @ bot_bot_set_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_919_SN__on__induct__acc__style,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,A2: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( ! [X2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ X2 @ bot_bot_set_term_a_b ) )
=> ( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y7 ) @ R5 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_920_SN__on__induct__acc__style,axiom,
! [R5: set_Pr1261947904930325089at_nat,A2: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat] :
( ( abstract_SN_on_nat @ R5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y7 ) @ R5 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X2 ) ) )
=> ( P2 @ A2 ) ) ) ).
% SN_on_induct_acc_style
thf(fact_921_step__preserves__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% step_preserves_SN_on
thf(fact_922_step__preserves__SN__on,axiom,
! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% step_preserves_SN_on
thf(fact_923_SN__on__imp__on__minimal,axiom,
! [R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) )
=> ! [Q2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ Q2 )
=> ? [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ Q2 )
& ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y7 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_924_SN__on__imp__on__minimal,axiom,
! [R: set_Pr4386577575007340137rm_a_b,X: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) )
=> ! [Q2: set_term_a_b] :
( ( member_term_a_b @ X @ Q2 )
=> ? [X2: term_a_b] :
( ( member_term_a_b @ X2 @ Q2 )
& ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y7 ) @ R )
=> ~ ( member_term_a_b @ Y7 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_925_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1261947904930325089at_nat,X: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ! [Q2: set_nat] :
( ( member_nat @ X @ Q2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ Q2 )
& ! [Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y7 ) @ R )
=> ~ ( member_nat @ Y7 @ Q2 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_926_step__reflects__SN__on,axiom,
! [A2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [B: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ R )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B @ bot_bot_set_term_a_b ) ) )
=> ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% step_reflects_SN_on
thf(fact_927_step__reflects__SN__on,axiom,
! [A2: nat,R: set_Pr1261947904930325089at_nat] :
( ! [B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% step_reflects_SN_on
thf(fact_928_SN__on__induct_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,S: term_a_b,P2: term_a_b > $o] :
( ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ S @ bot_bot_set_term_a_b ) )
=> ( ! [T3: term_a_b] :
( ! [U4: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T3 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T3 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_929_SN__on__induct_H,axiom,
! [R5: set_Pr1261947904930325089at_nat,S: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R5 @ ( insert_nat @ S @ bot_bot_set_nat ) )
=> ( ! [T3: nat] :
( ! [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T3 @ U4 ) @ R5 )
=> ( P2 @ U4 ) )
=> ( P2 @ T3 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_930_SN__I,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ! [A4: term_a_b] : ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ A4 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b ) ) ).
% SN_I
thf(fact_931_SN__I,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [A4: nat] : ( abstract_SN_on_nat @ A @ ( insert_nat @ A4 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A @ top_top_set_nat ) ) ).
% SN_I
thf(fact_932_SN__I,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ! [A4: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b ) )
=> ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ) ).
% SN_I
thf(fact_933_SN__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
= ( ! [X4: term_a_b] : ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ X4 @ bot_bot_set_term_a_b ) ) ) ) ).
% SN_def
thf(fact_934_SN__def,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
= ( ! [X4: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% SN_def
thf(fact_935_SN__def,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ top_to1314267278846557113rm_a_b )
= ( ! [X4: produc357393685978478089rm_a_b] : ( abstra2398554102055911763rm_a_b @ R @ ( insert7009541432154983385rm_a_b @ X4 @ bot_bo197521221353338581rm_a_b ) ) ) ) ).
% SN_def
thf(fact_936_NF__imp__SN__on,axiom,
! [A2: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member5869715511025134514rm_a_b @ A2 @ ( abstra1902783451776770197rm_a_b @ R5 ) )
=> ( abstra2398554102055911763rm_a_b @ R5 @ ( insert7009541432154983385rm_a_b @ A2 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_937_NF__imp__SN__on,axiom,
! [A2: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member_term_a_b @ A2 @ ( abstract_NF_term_a_b @ R5 ) )
=> ( abstra4720023341729745482rm_a_b @ R5 @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ).
% NF_imp_SN_on
thf(fact_938_NF__imp__SN__on,axiom,
! [A2: nat,R5: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( abstract_NF_nat @ R5 ) )
=> ( abstract_SN_on_nat @ R5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% NF_imp_SN_on
thf(fact_939_steps__preserve__SN__on,axiom,
! [A2: term_a_b,B2: term_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ A ) )
=> ( ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ( abstra4720023341729745482rm_a_b @ A @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_940_steps__preserve__SN__on,axiom,
! [A2: nat,B2: nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ A ) )
=> ( ( abstract_SN_on_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_941_steps__reflect__SN__on,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B2: term_a_b,A2: term_a_b] :
( ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( transi7742714808557438673rm_a_b @ R ) )
=> ~ ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_942_steps__reflect__SN__on,axiom,
! [R: set_Pr1261947904930325089at_nat,B2: nat,A2: nat] :
( ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_943_SN__on__imp__normalizability,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ ( insert_term_a_b @ A2 @ bot_bot_set_term_a_b ) )
=> ? [B: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ ( abstra2972651231392952684rm_a_b @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_944_SN__on__imp__normalizability,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ? [B: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( abstra7312431243108365096ty_nat @ R ) ) ) ).
% SN_on_imp_normalizability
thf(fact_945_insert__partition,axiom,
! [X: set_nat,F4: set_set_nat] :
( ~ ( member_set_nat @ X @ F4 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( insert_set_nat @ X @ F4 ) )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ ( insert_set_nat @ X @ F4 ) )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf_set_nat @ X2 @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F4 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_946_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_947_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_948_min__ext__compat,axiom,
! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R5 @ S4 ) @ R5 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R5 ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S4 ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R5 ) ) ) ).
% min_ext_compat
thf(fact_949_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_950_is__singletonI_H,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_sin332545968047684349rm_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_951_is__singletonI_H,axiom,
! [A: set_term_a_b] :
( ( A != bot_bot_set_term_a_b )
=> ( ! [X2: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ( member_term_a_b @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_sin5099058466283458592rm_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_952_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_953_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_954_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X4: nat] :
( A6
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_955_max__ext__compat,axiom,
! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R5 @ S4 ) @ R5 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R5 ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S4 ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R5 ) ) ) ).
% max_ext_compat
thf(fact_956_Field__insert,axiom,
! [A2: term_a_b,B2: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( field_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) )
= ( sup_sup_set_term_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) ) ) ).
% Field_insert
thf(fact_957_Field__insert,axiom,
! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_958_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_959_FieldI1,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
=> ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_960_FieldI1,axiom,
! [I: nat,J: nat,R5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R5 )
=> ( member_nat @ I @ ( field_nat @ R5 ) ) ) ).
% FieldI1
thf(fact_961_FieldI1,axiom,
! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
=> ( member_term_a_b @ I @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI1
thf(fact_962_FieldI2,axiom,
! [I: produc357393685978478089rm_a_b,J: produc357393685978478089rm_a_b,R5: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R5 )
=> ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_963_FieldI2,axiom,
! [I: nat,J: nat,R5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R5 )
=> ( member_nat @ J @ ( field_nat @ R5 ) ) ) ).
% FieldI2
thf(fact_964_FieldI2,axiom,
! [I: term_a_b,J: term_a_b,R5: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R5 )
=> ( member_term_a_b @ J @ ( field_term_a_b @ R5 ) ) ) ).
% FieldI2
thf(fact_965_remove__def,axiom,
( remove_nat
= ( ^ [X4: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_966_member__remove,axiom,
! [X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ ( remove8523906646656265582rm_a_b @ Y @ A ) )
= ( ( member5869715511025134514rm_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_967_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_968_member__remove,axiom,
! [X: term_a_b,Y: term_a_b,A: set_term_a_b] :
( ( member_term_a_b @ X @ ( remove_term_a_b @ Y @ A ) )
= ( ( member_term_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_969_acyclic__insert,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( transi5314701259734769157rm_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
= ( ( transi5314701259734769157rm_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% acyclic_insert
thf(fact_970_acyclicI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ ( transi7922773638565587891rm_a_b @ R ) )
=> ( transi5314701259734769157rm_a_b @ R ) ) ).
% acyclicI
thf(fact_971_acyclic__def,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ X4 ) @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_972_SN__imp__acyclic,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R5 @ top_top_set_term_a_b )
=> ( transi5314701259734769157rm_a_b @ R5 ) ) ).
% SN_imp_acyclic
thf(fact_973_SN__imp__acyclic,axiom,
! [R5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R5 @ top_top_set_nat )
=> ( transi3139058337892597825ic_nat @ R5 ) ) ).
% SN_imp_acyclic
thf(fact_974_SN__imp__acyclic,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R5 @ top_to1314267278846557113rm_a_b )
=> ( transi823393136465916248rm_a_b @ R5 ) ) ).
% SN_imp_acyclic
thf(fact_975_acyclic__irrefl,axiom,
( transi5314701259734769157rm_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] : ( irrefl_on_term_a_b @ top_top_set_term_a_b @ ( transi7922773638565587891rm_a_b @ R4 ) ) ) ) ).
% acyclic_irrefl
thf(fact_976_acyclic__irrefl,axiom,
( transi3139058337892597825ic_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] : ( irrefl_on_nat @ top_top_set_nat @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).
% acyclic_irrefl
thf(fact_977_acyclic__irrefl,axiom,
( transi823393136465916248rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] : ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ ( transi9211502839322181930rm_a_b @ R4 ) ) ) ) ).
% acyclic_irrefl
thf(fact_978_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_979_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat,Phi: nat > nat > $o] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
=> ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( A2 = B2 )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_980_asym__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_981_asym__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( asym_on_nat @ top_top_set_nat @ R_B )
=> ( asym_o316262460646556663_b_nat @ top_to1329314642032814325_b_nat @ ( lex_pr3461217179711548565_b_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_982_asym__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( asym_o6083219562588744120rm_a_b @ top_to3635194522128982992rm_a_b @ ( lex_pr5735081776509173764rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_983_asym__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr4386577575007340137rm_a_b] :
( ( asym_on_nat @ top_top_set_nat @ R_A )
=> ( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( asym_o8054106581205124983rm_a_b @ top_to8394164535439305845rm_a_b @ ( lex_pr2499763052546576405rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_984_asym__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( asym_on_nat @ top_top_set_nat @ R_A )
=> ( ( asym_on_nat @ top_top_set_nat @ R_B )
=> ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_985_asym__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr2972776593051762503rm_a_b] :
( ( asym_on_nat @ top_top_set_nat @ R_A )
=> ( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( asym_o7558514146396566396rm_a_b @ top_to5613645074002683412rm_a_b @ ( lex_pr5733628728943162184rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_986_asym__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( asym_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( asym_o5349708740260273946rm_a_b @ top_to5343659064632936498rm_a_b @ ( lex_pr7567299705543466414rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_987_asym__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( asym_on_nat @ top_top_set_nat @ R_B )
=> ( asym_o3458304777164054614_b_nat @ top_to8128465324599887342_b_nat @ ( lex_pr3675074321942177898_b_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_988_asym__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( asym_o346804095348748697rm_a_b @ top_to1302229308806327447rm_a_b @ ( lex_pr3939907742592656111rm_a_b @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_989_in__lex__prod,axiom,
! [A2: term_a_b,B2: term_a_b,A3: term_a_b,B3: term_a_b,R: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( lex_pr6163557265797435481rm_a_b @ R @ S ) )
= ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_990_wo__rel_OTOTALS,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ! [X6: term_a_b] :
( ( member_term_a_b @ X6 @ ( field_term_a_b @ R ) )
=> ! [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Xa2 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa2 @ X6 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_991_wo__rel_Oin__notinI,axiom,
! [R: set_Pr2972776593051762503rm_a_b,J: produc357393685978478089rm_a_b,I: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member5869715511025134514rm_a_b @ I @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ J @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_992_wo__rel_Oin__notinI,axiom,
! [R: set_Pr1261947904930325089at_nat,J: nat,I: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member_nat @ I @ ( field_nat @ R ) )
=> ( ( member_nat @ J @ ( field_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_993_wo__rel_Oin__notinI,axiom,
! [R: set_Pr4386577575007340137rm_a_b,J: term_a_b,I: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ J @ I ) @ R )
| ( J = I ) )
=> ( ( member_term_a_b @ I @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ J @ ( field_term_a_b @ R ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ I @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_994_wo__rel_Owell__order__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X2: term_a_b] :
( ! [Y7: term_a_b] :
( ( ( Y7 != X2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X2 ) @ R ) )
=> ( P2 @ Y7 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A2 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_995_wo__rel_Ocases__Total,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b,Phi: term_a_b > term_a_b > $o] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) @ ( field_term_a_b @ R ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_996_wo__rel_Ocases__Total,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat,Phi: nat > nat > $o] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ R )
=> ( Phi @ A2 @ B2 ) )
=> ( Phi @ A2 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_997_irrefl__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_998_irrefl__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
=> ( irrefl8120777048070186683_b_nat @ top_to1329314642032814325_b_nat @ ( lex_pr3461217179711548565_b_nat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_999_irrefl__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( irrefl8409159218339128820rm_a_b @ top_to3635194522128982992rm_a_b @ ( lex_pr5735081776509173764rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1000_irrefl__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr4386577575007340137rm_a_b] :
( ( irrefl_on_nat @ top_top_set_nat @ R_A )
=> ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( irrefl6635249131773979195rm_a_b @ top_to8394164535439305845rm_a_b @ ( lex_pr2499763052546576405rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1001_irrefl__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( irrefl_on_nat @ top_top_set_nat @ R_A )
=> ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
=> ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1002_irrefl__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr2972776593051762503rm_a_b] :
( ( irrefl_on_nat @ top_top_set_nat @ R_A )
=> ( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( irrefl1004993053404085688rm_a_b @ top_to5613645074002683412rm_a_b @ ( lex_pr5733628728943162184rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1003_irrefl__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( irrefl7675648396010658646rm_a_b @ top_to5343659064632936498rm_a_b @ ( lex_pr7567299705543466414rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1004_irrefl__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
=> ( irrefl6128155721026349714_b_nat @ top_to8128465324599887342_b_nat @ ( lex_pr3675074321942177898_b_nat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1005_irrefl__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( irrefl5705139877864982621rm_a_b @ top_to1302229308806327447rm_a_b @ ( lex_pr3939907742592656111rm_a_b @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_1006_well__order__induct__imp,axiom,
! [R: set_Pr2972776593051762503rm_a_b,P2: produc357393685978478089rm_a_b > $o,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ! [Y7: produc357393685978478089rm_a_b] :
( ( ( Y7 != X2 )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ X2 ) @ R ) )
=> ( ( member5869715511025134514rm_a_b @ Y7 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P2 @ Y7 ) ) )
=> ( ( member5869715511025134514rm_a_b @ X2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P2 @ X2 ) ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( P2 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1007_well__order__induct__imp,axiom,
! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ! [X2: nat] :
( ! [Y7: nat] :
( ( ( Y7 != X2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X2 ) @ R ) )
=> ( ( member_nat @ Y7 @ ( field_nat @ R ) )
=> ( P2 @ Y7 ) ) )
=> ( ( member_nat @ X2 @ ( field_nat @ R ) )
=> ( P2 @ X2 ) ) )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( P2 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1008_well__order__induct__imp,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ! [X2: term_a_b] :
( ! [Y7: term_a_b] :
( ( ( Y7 != X2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X2 ) @ R ) )
=> ( ( member_term_a_b @ Y7 @ ( field_term_a_b @ R ) )
=> ( P2 @ Y7 ) ) )
=> ( ( member_term_a_b @ X2 @ ( field_term_a_b @ R ) )
=> ( P2 @ X2 ) ) )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( P2 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1009_wo__rel_Oequals__minim,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B6: set_Pr4386577575007340137rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ A2 @ B6 )
=> ( ! [B: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ B @ B6 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B ) @ R ) )
=> ( A2
= ( bNF_We1338740440619677141rm_a_b @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1010_wo__rel_Oequals__minim,axiom,
! [R: set_Pr1261947904930325089at_nat,B6: set_nat,A2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
=> ( ( member_nat @ A2 @ B6 )
=> ( ! [B: nat] :
( ( member_nat @ B @ B6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R ) )
=> ( A2
= ( bNF_We3275124272252294916im_nat @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1011_wo__rel_Oequals__minim,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B6: set_term_a_b,A2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B6 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ A2 @ B6 )
=> ( ! [B: term_a_b] :
( ( member_term_a_b @ B @ B6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B ) @ R ) )
=> ( A2
= ( bNF_We6258903063523145544rm_a_b @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1012_ssubst__Pair__rhs,axiom,
! [R: term_a_b,S: term_a_b,R5: set_Pr4386577575007340137rm_a_b,S7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S ) @ R5 )
=> ( ( S7 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R @ S7 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1013_wo__rel_Ominim__least,axiom,
! [R: set_Pr2972776593051762503rm_a_b,B6: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( ord_le118470702582115849rm_a_b @ B6 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ B6 )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( bNF_We1338740440619677141rm_a_b @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1014_wo__rel_Ominim__least,axiom,
! [R: set_Pr1261947904930325089at_nat,B6: set_nat,B2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( ord_less_eq_set_nat @ B6 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ B6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1015_wo__rel_Ominim__least,axiom,
! [R: set_Pr4386577575007340137rm_a_b,B6: set_term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ord_le2705286416250468010rm_a_b @ B6 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ B6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( bNF_We6258903063523145544rm_a_b @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1016_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) @ ( insert7009541432154983385rm_a_b @ A2 @ ( insert7009541432154983385rm_a_b @ B2 @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1017_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member_term_a_b @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) @ ( insert_term_a_b @ A2 @ ( insert_term_a_b @ B2 @ bot_bot_set_term_a_b ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1018_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ ( field_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 ) ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 ) ) @ R )
& ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 ) @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1019_refl__on__singleton,axiom,
! [X: term_a_b] : ( refl_on_term_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% refl_on_singleton
thf(fact_1020_refl__on__singleton,axiom,
! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_1021_refl__rtrancl,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( transi7742714808557438673rm_a_b @ R ) ) ).
% refl_rtrancl
thf(fact_1022_refl__rtrancl,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( refl_on_nat @ top_top_set_nat @ ( transi2905341329935302413cl_nat @ R ) ) ).
% refl_rtrancl
thf(fact_1023_refl__rtrancl,axiom,
! [R: set_Pr2972776593051762503rm_a_b] : ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ ( transi2615809358984392588rm_a_b @ R ) ) ).
% refl_rtrancl
thf(fact_1024_refl__Id,axiom,
refl_on_term_a_b @ top_top_set_term_a_b @ id_term_a_b ).
% refl_Id
thf(fact_1025_refl__Id,axiom,
refl_on_nat @ top_top_set_nat @ id_nat ).
% refl_Id
thf(fact_1026_refl__Id,axiom,
refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ id_Pro3488709451141449061rm_a_b ).
% refl_Id
thf(fact_1027_reflI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ X2 ) @ R )
=> ( refl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% reflI
thf(fact_1028_reflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ( refl_on_nat @ top_top_set_nat @ R ) ) ).
% reflI
thf(fact_1029_reflI,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b] : ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ X2 ) @ R )
=> ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% reflI
thf(fact_1030_reflD,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1031_reflD,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( refl_on_nat @ top_top_set_nat @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1032_reflD,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ R )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1033_refl__onD2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_1034_refl__onD2,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member_nat @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_1035_refl__onD2,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member_term_a_b @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_1036_refl__onD1,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Y: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X @ Y ) @ R )
=> ( member5869715511025134514rm_a_b @ X @ A ) ) ) ).
% refl_onD1
thf(fact_1037_refl__onD1,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member_nat @ X @ A ) ) ) ).
% refl_onD1
thf(fact_1038_refl__onD1,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,X: term_a_b,Y: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R )
=> ( member_term_a_b @ X @ A ) ) ) ).
% refl_onD1
thf(fact_1039_refl__onD,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
=> ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1040_refl__onD,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,A2: nat] :
( ( refl_on_nat @ A @ R )
=> ( ( member_nat @ A2 @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1041_refl__onD,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member_term_a_b @ A2 @ A )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1042_refl__on__empty,axiom,
refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% refl_on_empty
thf(fact_1043_refl__lex__prod,axiom,
! [R_B: set_Pr4386577575007340137rm_a_b,R_A: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ).
% refl_lex_prod
thf(fact_1044_wo__rel_Omax2__def,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 ) )
& ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1045_refl__O__iff,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( refl_on_term_a_b @ top_top_set_term_a_b @ A )
= ( ord_le118470702582115849rm_a_b @ id_term_a_b @ A ) ) ).
% refl_O_iff
thf(fact_1046_refl__O__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( refl_on_nat @ top_top_set_nat @ A )
= ( ord_le3146513528884898305at_nat @ id_nat @ A ) ) ).
% refl_O_iff
thf(fact_1047_refl__O__iff,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ A )
= ( ord_le2593958535671617255rm_a_b @ id_Pro3488709451141449061rm_a_b @ A ) ) ).
% refl_O_iff
thf(fact_1048_refl__reflcl,axiom,
! [R: set_Pr4386577575007340137rm_a_b] : ( refl_on_term_a_b @ top_top_set_term_a_b @ ( sup_su6776935440552674877rm_a_b @ R @ id_term_a_b ) ) ).
% refl_reflcl
thf(fact_1049_refl__reflcl,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( refl_on_nat @ top_top_set_nat @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) ) ).
% refl_reflcl
thf(fact_1050_refl__reflcl,axiom,
! [R: set_Pr2972776593051762503rm_a_b] : ( refl_o5344404488616963464rm_a_b @ top_to1314267278846557113rm_a_b @ ( sup_su1845815533690797339rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ).
% refl_reflcl
thf(fact_1051_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1052_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ ( field_nat @ R ) )
=> ( ( ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 )
= A2 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1053_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= A2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1054_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1055_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ ( field_nat @ R ) )
=> ( ( ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 )
= B2 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1056_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 )
= B2 )
= ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1057_wo__rel_Omax2__greater,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( bNF_We1688910813892497397rm_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ ( bNF_We1369222903935127127rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1058_wo__rel_Omax2__greater,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( bNF_We3818239936649020644el_nat @ R )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ ( field_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 ) ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ ( bNF_We2380939065524664066x2_nat @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1059_wo__rel_Omax2__greater,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( bNF_We251836928773027112rm_a_b @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ ( bNF_We4618759378340593734rm_a_b @ R @ A2 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1060_refl__on__domain,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( refl_o5344404488616963464rm_a_b @ A @ R )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ A )
& ( member5869715511025134514rm_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1061_refl__on__domain,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( refl_on_nat @ A @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
=> ( ( member_nat @ A2 @ A )
& ( member_nat @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1062_refl__on__domain,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( refl_on_term_a_b @ A @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
=> ( ( member_term_a_b @ A2 @ A )
& ( member_term_a_b @ B2 @ A ) ) ) ) ).
% refl_on_domain
thf(fact_1063_Total__subset__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( ord_le118470702582115849rm_a_b @ R @ id_term_a_b )
=> ( ( R = bot_bo197521221353338581rm_a_b )
| ? [A4: term_a_b] :
( R
= ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ A4 @ A4 ) @ bot_bo197521221353338581rm_a_b ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1064_linear__order__on__singleton,axiom,
! [X: term_a_b] : ( order_5388802246213473311rm_a_b @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ bot_bo197521221353338581rm_a_b ) ) ).
% linear_order_on_singleton
thf(fact_1065_linear__order__on__singleton,axiom,
! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1066_total__on__singleton,axiom,
! [X: nat,R: set_Pr1261947904930325089at_nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R ) ).
% total_on_singleton
thf(fact_1067_total__on__empty,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R ) ).
% total_on_empty
thf(fact_1068_total__on__def,axiom,
( total_on_term_a_b
= ( ^ [A6: set_term_a_b,R4: set_Pr4386577575007340137rm_a_b] :
! [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
=> ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R4 )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1069_total__onI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ( member5869715511025134514rm_a_b @ Y3 @ A )
=> ( ( X2 != Y3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R ) ) ) ) )
=> ( total_4010809703584253837rm_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1070_total__onI,axiom,
! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( X2 != Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R ) ) ) ) )
=> ( total_on_nat @ A @ R ) ) ).
% total_onI
thf(fact_1071_total__onI,axiom,
! [A: set_term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ( member_term_a_b @ Y3 @ A )
=> ( ( X2 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R ) ) ) ) )
=> ( total_on_term_a_b @ A @ R ) ) ).
% total_onI
thf(fact_1072_totalI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Y3: term_a_b] :
( ( X2 != Y3 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X2 ) @ R ) ) )
=> ( total_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% totalI
thf(fact_1073_totalI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R ) ) )
=> ( total_on_nat @ top_top_set_nat @ R ) ) ).
% totalI
thf(fact_1074_totalI,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Y3: produc357393685978478089rm_a_b] :
( ( X2 != Y3 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X2 @ Y3 ) @ R )
| ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ X2 ) @ R ) ) )
=> ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% totalI
thf(fact_1075_total__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( total_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( total_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ ( lex_pr6163557265797435481rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1076_total__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( total_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( total_on_nat @ top_top_set_nat @ R_B )
=> ( total_2322242646682082889_b_nat @ top_to1329314642032814325_b_nat @ ( lex_pr3461217179711548565_b_nat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1077_total__lex__prod,axiom,
! [R_A: set_Pr4386577575007340137rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( total_on_term_a_b @ top_top_set_term_a_b @ R_A )
=> ( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( total_3815809079797891814rm_a_b @ top_to3635194522128982992rm_a_b @ ( lex_pr5735081776509173764rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1078_total__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr4386577575007340137rm_a_b] :
( ( total_on_nat @ top_top_set_nat @ R_A )
=> ( ( total_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( total_836714730385875401rm_a_b @ top_to8394164535439305845rm_a_b @ ( lex_pr2499763052546576405rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1079_total__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( total_on_nat @ top_top_set_nat @ R_A )
=> ( ( total_on_nat @ top_top_set_nat @ R_B )
=> ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1080_total__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr2972776593051762503rm_a_b] :
( ( total_on_nat @ top_top_set_nat @ R_A )
=> ( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( total_7403620232476326826rm_a_b @ top_to5613645074002683412rm_a_b @ ( lex_pr5733628728943162184rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1081_total__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr4386577575007340137rm_a_b] :
( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( total_on_term_a_b @ top_top_set_term_a_b @ R_B )
=> ( total_3082298257469421640rm_a_b @ top_to5343659064632936498rm_a_b @ ( lex_pr7567299705543466414rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1082_total__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr1261947904930325089at_nat] :
( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( total_on_nat @ top_top_set_nat @ R_B )
=> ( total_3303410863243815044_b_nat @ top_to8128465324599887342_b_nat @ ( lex_pr3675074321942177898_b_nat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1083_total__lex__prod,axiom,
! [R_A: set_Pr2972776593051762503rm_a_b,R_B: set_Pr2972776593051762503rm_a_b] :
( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_A )
=> ( ( total_4010809703584253837rm_a_b @ top_to1314267278846557113rm_a_b @ R_B )
=> ( total_5213408123548685803rm_a_b @ top_to1302229308806327447rm_a_b @ ( lex_pr3939907742592656111rm_a_b @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_1084_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2972776593051762503rm_a_b,A2: produc357393685978478089rm_a_b,B2: produc357393685978478089rm_a_b] :
( ( order_681589137112254398rm_a_b @ ( field_6884932134483023318rm_a_b @ R ) @ R )
=> ( ( member5869715511025134514rm_a_b @ A2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ B2 @ ( field_6884932134483023318rm_a_b @ R ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ B2 @ A2 ) @ ( minus_45395518312058030rm_a_b @ R @ id_Pro3488709451141449061rm_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1085_Linear__order__in__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: nat,B2: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A2 @ ( field_nat @ R ) )
=> ( ( member_nat @ B2 @ ( field_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1086_Linear__order__in__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,B2: term_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( member_term_a_b @ A2 @ ( field_term_a_b @ R ) )
=> ( ( member_term_a_b @ B2 @ ( field_term_a_b @ R ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ B2 ) @ R )
= ( ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B2 @ A2 ) @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1087_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( order_5388802246213473311rm_a_b @ ( field_term_a_b @ R ) @ R )
=> ( ( wf_term_a_b @ ( minus_5192120951422937424rm_a_b @ R @ id_term_a_b ) )
= ( ! [A6: set_term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R ) )
=> ( ( A6 != bot_bot_set_term_a_b )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1088_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat ) )
= ( ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
=> ( ( A6 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1089_prod__set__simps_I2_J,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( basic_6690576665378965359rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
= ( insert_term_a_b @ Y @ bot_bot_set_term_a_b ) ) ).
% prod_set_simps(2)
thf(fact_1090_wf__insert,axiom,
! [Y: term_a_b,X: term_a_b,R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( insert7009541432154983385rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ X ) @ R ) )
= ( ( wf_term_a_b @ R )
& ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ ( transi7742714808557438673rm_a_b @ R ) ) ) ) ).
% wf_insert
thf(fact_1091_wf__imp__asym,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R )
=> ( asym_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% wf_imp_asym
thf(fact_1092_wf__imp__asym,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R )
=> ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% wf_imp_asym
thf(fact_1093_wf__imp__asym,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R )
=> ( asym_o1443355727950249531rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% wf_imp_asym
thf(fact_1094_wfE__min_H,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( Q != bot_bo197521221353338581rm_a_b )
=> ~ ! [Z: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z @ Q )
=> ~ ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1095_wfE__min_H,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( Q != bot_bot_set_term_a_b )
=> ~ ! [Z: term_a_b] :
( ( member_term_a_b @ Z @ Q )
=> ~ ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z ) @ R5 )
=> ~ ( member_term_a_b @ Y7 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1096_wfE__min_H,axiom,
! [R5: set_Pr1261947904930325089at_nat,Q: set_nat] :
( ( wf_nat @ R5 )
=> ( ( Q != bot_bot_set_nat )
=> ~ ! [Z: nat] :
( ( member_nat @ Z @ Q )
=> ~ ! [Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Z ) @ R5 )
=> ~ ( member_nat @ Y7 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1097_wf__if__convertible__to__wf,axiom,
! [S: set_Pr4386577575007340137rm_a_b,R: set_Pr4386577575007340137rm_a_b,F3: term_a_b > term_a_b] :
( ( wf_term_a_b @ S )
=> ( ! [X2: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X2 @ Y3 ) @ R )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ X2 ) @ ( F3 @ Y3 ) ) @ S ) )
=> ( wf_term_a_b @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1098_wf__induct__rule,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X2: term_a_b] :
( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X2 ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_1099_wf__eq__minimal,axiom,
( wf_Pro2335863617654816626rm_a_b
= ( ^ [R4: set_Pr2972776593051762503rm_a_b] :
! [Q3: set_Pr4386577575007340137rm_a_b] :
( ? [X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ Q3 )
=> ? [X4: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X4 @ Q3 )
& ! [Y5: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y5 @ X4 ) @ R4 )
=> ~ ( member5869715511025134514rm_a_b @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1100_wf__eq__minimal,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [Q3: set_nat] :
( ? [X4: nat] : ( member_nat @ X4 @ Q3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ Q3 )
& ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R4 )
=> ~ ( member_nat @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1101_wf__eq__minimal,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [Q3: set_term_a_b] :
( ? [X4: term_a_b] : ( member_term_a_b @ X4 @ Q3 )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ Q3 )
& ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
=> ~ ( member_term_a_b @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1102_wf__not__refl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% wf_not_refl
thf(fact_1103_wf__not__sym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A2 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_1104_wf__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ A2 ) @ R ) ) ).
% wf_irrefl
thf(fact_1105_wf__induct,axiom,
! [R: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o,A2: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ! [X2: term_a_b] :
( ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ X2 ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A2 ) ) ) ).
% wf_induct
thf(fact_1106_wf__asym,axiom,
! [R: set_Pr4386577575007340137rm_a_b,A2: term_a_b,X: term_a_b] :
( ( wf_term_a_b @ R )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A2 @ X ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ A2 ) @ R ) ) ) ).
% wf_asym
thf(fact_1107_wfUNIVI,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [P3: term_a_b > $o,X2: term_a_b] :
( ! [Xa2: term_a_b] :
( ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ Xa2 ) )
=> ( P3 @ X2 ) )
=> ( wf_term_a_b @ R ) ) ).
% wfUNIVI
thf(fact_1108_wfI__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b,Q4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ Q4 )
=> ? [Xa2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xa2 @ Q4 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q4 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1109_wfI__min,axiom,
! [R5: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Q4: set_nat] :
( ( member_nat @ X2 @ Q4 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ Q4 )
& ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_nat @ Y3 @ Q4 ) ) ) )
=> ( wf_nat @ R5 ) ) ).
% wfI_min
thf(fact_1110_wfI__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b,Q4: set_term_a_b] :
( ( member_term_a_b @ X2 @ Q4 )
=> ? [Xa2: term_a_b] :
( ( member_term_a_b @ Xa2 @ Q4 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Xa2 ) @ R5 )
=> ~ ( member_term_a_b @ Y3 @ Q4 ) ) ) )
=> ( wf_term_a_b @ R5 ) ) ).
% wfI_min
thf(fact_1111_wfE__min,axiom,
! [R5: set_Pr2972776593051762503rm_a_b,X: produc357393685978478089rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R5 )
=> ( ( member5869715511025134514rm_a_b @ X @ Q )
=> ~ ! [Z: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ Z @ Q )
=> ~ ! [Y7: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ Y7 @ Z ) @ R5 )
=> ~ ( member5869715511025134514rm_a_b @ Y7 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1112_wfE__min,axiom,
! [R5: set_Pr1261947904930325089at_nat,X: nat,Q: set_nat] :
( ( wf_nat @ R5 )
=> ( ( member_nat @ X @ Q )
=> ~ ! [Z: nat] :
( ( member_nat @ Z @ Q )
=> ~ ! [Y7: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Z ) @ R5 )
=> ~ ( member_nat @ Y7 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1113_wfE__min,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,X: term_a_b,Q: set_term_a_b] :
( ( wf_term_a_b @ R5 )
=> ( ( member_term_a_b @ X @ Q )
=> ~ ! [Z: term_a_b] :
( ( member_term_a_b @ Z @ Q )
=> ~ ! [Y7: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y7 @ Z ) @ R5 )
=> ~ ( member_term_a_b @ Y7 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1114_wf__def,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [P4: term_a_b > $o] :
( ! [X4: term_a_b] :
( ! [Y5: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X4 ) )
=> ! [X7: term_a_b] : ( P4 @ X7 ) ) ) ) ).
% wf_def
thf(fact_1115_minimal__imp__wf,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ! [Q4: set_Pr4386577575007340137rm_a_b] :
( ? [X2: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X2 @ Q4 )
=> ? [X6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X6 @ Q4 )
& ! [Y3: produc357393685978478089rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ X6 @ Y3 ) @ R )
=> ~ ( member5869715511025134514rm_a_b @ Y3 @ Q4 ) ) ) )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1116_minimal__imp__wf,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [Q4: set_nat] :
( ? [X2: nat] : ( member_nat @ X2 @ Q4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ Q4 )
& ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y3 ) @ R )
=> ~ ( member_nat @ Y3 @ Q4 ) ) ) )
=> ( wf_nat @ ( converse_nat_nat @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1117_minimal__imp__wf,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ! [Q4: set_term_a_b] :
( ? [X2: term_a_b] : ( member_term_a_b @ X2 @ Q4 )
=> ? [X6: term_a_b] :
( ( member_term_a_b @ X6 @ Q4 )
& ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y3 ) @ R )
=> ~ ( member_term_a_b @ Y3 @ Q4 ) ) ) )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% minimal_imp_wf
thf(fact_1118_wf__imp__irrefl,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ R )
=> ( irrefl_on_term_a_b @ top_top_set_term_a_b @ R ) ) ).
% wf_imp_irrefl
thf(fact_1119_wf__imp__irrefl,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R )
=> ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% wf_imp_irrefl
thf(fact_1120_wf__imp__irrefl,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ R )
=> ( irrefl6620156600162922239rm_a_b @ top_to1314267278846557113rm_a_b @ R ) ) ).
% wf_imp_irrefl
thf(fact_1121_wf__imp__SN,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ A ) )
=> ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b ) ) ).
% wf_imp_SN
thf(fact_1122_wf__imp__SN,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ ( converse_nat_nat @ A ) )
=> ( abstract_SN_on_nat @ A @ top_top_set_nat ) ) ).
% wf_imp_SN
thf(fact_1123_wf__imp__SN,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ A ) )
=> ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b ) ) ).
% wf_imp_SN
thf(fact_1124_SN__imp__wf,axiom,
! [R: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ R @ top_top_set_term_a_b )
=> ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ R ) ) ) ).
% SN_imp_wf
thf(fact_1125_SN__imp__wf,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( wf_nat @ ( converse_nat_nat @ R ) ) ) ).
% SN_imp_wf
thf(fact_1126_SN__imp__wf,axiom,
! [R: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ R @ top_to1314267278846557113rm_a_b )
=> ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ R ) ) ) ).
% SN_imp_wf
thf(fact_1127_SN__iff__wf,axiom,
! [A: set_Pr4386577575007340137rm_a_b] :
( ( abstra4720023341729745482rm_a_b @ A @ top_top_set_term_a_b )
= ( wf_term_a_b @ ( conver7608559976937845093rm_a_b @ A ) ) ) ).
% SN_iff_wf
thf(fact_1128_SN__iff__wf,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ A @ top_top_set_nat )
= ( wf_nat @ ( converse_nat_nat @ A ) ) ) ).
% SN_iff_wf
thf(fact_1129_SN__iff__wf,axiom,
! [A: set_Pr2972776593051762503rm_a_b] :
( ( abstra2398554102055911763rm_a_b @ A @ top_to1314267278846557113rm_a_b )
= ( wf_Pro2335863617654816626rm_a_b @ ( conver1575690562132235771rm_a_b @ A ) ) ) ).
% SN_iff_wf
thf(fact_1130_wf__eq__minimal2,axiom,
( wf_term_a_b
= ( ^ [R4: set_Pr4386577575007340137rm_a_b] :
! [A6: set_term_a_b] :
( ( ( ord_le2705286416250468010rm_a_b @ A6 @ ( field_term_a_b @ R4 ) )
& ( A6 != bot_bot_set_term_a_b ) )
=> ? [X4: term_a_b] :
( ( member_term_a_b @ X4 @ A6 )
& ! [Y5: term_a_b] :
( ( member_term_a_b @ Y5 @ A6 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1131_wf__eq__minimal2,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [A6: set_nat] :
( ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R4 ) )
& ( A6 != bot_bot_set_nat ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1132_prod__set__simps_I1_J,axiom,
! [X: term_a_b,Y: term_a_b] :
( ( basic_3255686707823303985rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) )
= ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) ) ).
% prod_set_simps(1)
thf(fact_1133_cSup__inter__less__eq,axiom,
! [A: set_nat,B6: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( ( inf_inf_set_nat @ A @ B6 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A @ B6 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1134_bdd__above_OI,axiom,
! [A: set_nat,M3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1135_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_1136_bdd__above__insert,axiom,
! [A2: nat,A: set_nat] :
( ( condit2214826472909112428ve_nat @ ( insert_nat @ A2 @ A ) )
= ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above_insert
thf(fact_1137_bdd__above__Iic,axiom,
! [B2: nat] : ( condit2214826472909112428ve_nat @ ( set_ord_atMost_nat @ B2 ) ) ).
% bdd_above_Iic
thf(fact_1138_cSup__upper,axiom,
! [X: nat,X3: set_nat] :
( ( member_nat @ X @ X3 )
=> ( ( condit2214826472909112428ve_nat @ X3 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X3 ) ) ) ) ).
% cSup_upper
thf(fact_1139_cSup__upper2,axiom,
! [X: nat,X3: set_nat,Y: nat] :
( ( member_nat @ X @ X3 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X3 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X3 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1140_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M4: nat] :
~ ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_1141_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A6: set_nat] :
? [M5: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1142_cSup__mono,axiom,
! [B6: set_nat,A: set_nat] :
( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ! [B: nat] :
( ( member_nat @ B @ B6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ B @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B6 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1143_cSup__le__iff,axiom,
! [S4: set_nat,A2: nat] :
( ( S4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S4 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S4 ) @ A2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ S4 )
=> ( ord_less_eq_nat @ X4 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1144_cSup__subset__mono,axiom,
! [A: set_nat,B6: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1145_cSup__insert,axiom,
! [X3: set_nat,A2: nat] :
( ( X3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X3 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X3 ) )
= ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X3 ) ) ) ) ) ).
% cSup_insert
thf(fact_1146_cSup__insert__If,axiom,
! [X3: set_nat,A2: nat] :
( ( condit2214826472909112428ve_nat @ X3 )
=> ( ( ( X3 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X3 ) )
= A2 ) )
& ( ( X3 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X3 ) )
= ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X3 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1147_cSup__union__distrib,axiom,
! [A: set_nat,B6: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A @ B6 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1148_cSUP__union,axiom,
! [A: set_nat,F3: nat > nat,B6: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ B6 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ ( sup_sup_set_nat @ A @ B6 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1149_cSUP__insert,axiom,
! [A: set_nat,F3: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ ( insert_nat @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F3 @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_1150_image__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_3126328586747194693rm_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1151_image__eqI,axiom,
! [B2: nat,F3: produc357393685978478089rm_a_b > nat,X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member_nat @ B2 @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1152_image__eqI,axiom,
! [B2: term_a_b,F3: produc357393685978478089rm_a_b > term_a_b,X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member_term_a_b @ B2 @ ( image_2002304162058146776rm_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1153_image__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: nat > produc357393685978478089rm_a_b,X: nat,A: set_nat] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_7842151682427799154rm_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1154_image__eqI,axiom,
! [B2: nat,F3: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1155_image__eqI,axiom,
! [B2: term_a_b,F3: nat > term_a_b,X: nat,A: set_nat] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_term_a_b @ B2 @ ( image_nat_term_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1156_image__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: term_a_b > produc357393685978478089rm_a_b,X: term_a_b,A: set_term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_term_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_170086233023854126rm_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1157_image__eqI,axiom,
! [B2: nat,F3: term_a_b > nat,X: term_a_b,A: set_term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_term_a_b @ X @ A )
=> ( member_nat @ B2 @ ( image_term_a_b_nat @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1158_image__eqI,axiom,
! [B2: term_a_b,F3: term_a_b > term_a_b,X: term_a_b,A: set_term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( ( member_term_a_b @ X @ A )
=> ( member_term_a_b @ B2 @ ( image_3079262707727038895rm_a_b @ F3 @ A ) ) ) ) ).
% image_eqI
thf(fact_1159_image__empty,axiom,
! [F3: nat > nat] :
( ( image_nat_nat @ F3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1160_empty__is__image,axiom,
! [F3: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1161_image__is__empty,axiom,
! [F3: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F3 @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1162_image__insert,axiom,
! [F3: nat > nat,A2: nat,B6: set_nat] :
( ( image_nat_nat @ F3 @ ( insert_nat @ A2 @ B6 ) )
= ( insert_nat @ ( F3 @ A2 ) @ ( image_nat_nat @ F3 @ B6 ) ) ) ).
% image_insert
thf(fact_1163_insert__image,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( insert_nat @ ( F3 @ X ) @ ( image_5783597275426814868_b_nat @ F3 @ A ) )
= ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_1164_insert__image,axiom,
! [X: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A ) )
= ( image_nat_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_1165_insert__image,axiom,
! [X: term_a_b,A: set_term_a_b,F3: term_a_b > nat] :
( ( member_term_a_b @ X @ A )
=> ( ( insert_nat @ ( F3 @ X ) @ ( image_term_a_b_nat @ F3 @ A ) )
= ( image_term_a_b_nat @ F3 @ A ) ) ) ).
% insert_image
thf(fact_1166_bdd__above_OI2,axiom,
! [A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat,M3: nat] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( condit2214826472909112428ve_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ).
% bdd_above.I2
thf(fact_1167_bdd__above_OI2,axiom,
! [A: set_nat,F3: nat > nat,M3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) ) ) ).
% bdd_above.I2
thf(fact_1168_bdd__above_OI2,axiom,
! [A: set_term_a_b,F3: term_a_b > nat,M3: nat] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( condit2214826472909112428ve_nat @ ( image_term_a_b_nat @ F3 @ A ) ) ) ).
% bdd_above.I2
thf(fact_1169_Inf_OINF__cong,axiom,
! [A: set_nat,B6: set_nat,C2: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
( ( A = B6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( C2 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D3 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1170_Sup_OSUP__cong,axiom,
! [A: set_nat,B6: set_nat,C2: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
( ( A = B6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( C2 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D3 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1171_imageI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_3126328586747194693rm_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1172_imageI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_1173_imageI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > term_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( member_term_a_b @ ( F3 @ X ) @ ( image_2002304162058146776rm_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1174_imageI,axiom,
! [X: nat,A: set_nat,F3: nat > produc357393685978478089rm_a_b] :
( ( member_nat @ X @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_7842151682427799154rm_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1175_imageI,axiom,
! [X: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_1176_imageI,axiom,
! [X: nat,A: set_nat,F3: nat > term_a_b] :
( ( member_nat @ X @ A )
=> ( member_term_a_b @ ( F3 @ X ) @ ( image_nat_term_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1177_imageI,axiom,
! [X: term_a_b,A: set_term_a_b,F3: term_a_b > produc357393685978478089rm_a_b] :
( ( member_term_a_b @ X @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_170086233023854126rm_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1178_imageI,axiom,
! [X: term_a_b,A: set_term_a_b,F3: term_a_b > nat] :
( ( member_term_a_b @ X @ A )
=> ( member_nat @ ( F3 @ X ) @ ( image_term_a_b_nat @ F3 @ A ) ) ) ).
% imageI
thf(fact_1179_imageI,axiom,
! [X: term_a_b,A: set_term_a_b,F3: term_a_b > term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( member_term_a_b @ ( F3 @ X ) @ ( image_3079262707727038895rm_a_b @ F3 @ A ) ) ) ).
% imageI
thf(fact_1180_image__iff,axiom,
! [Z2: nat,F3: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F3 @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z2
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_1181_bex__imageD,axiom,
! [F3: nat > nat,A: set_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F3 @ A ) )
& ( P2 @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P2 @ ( F3 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_1182_image__cong,axiom,
! [M3: set_nat,N4: set_nat,F3: nat > nat,G: nat > nat] :
( ( M3 = N4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ( F3 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F3 @ M3 )
= ( image_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_1183_ball__imageD,axiom,
! [F3: nat > nat,A: set_nat,P2: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F3 @ A ) )
=> ( P2 @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P2 @ ( F3 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_1184_rev__image__eqI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: produc357393685978478089rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_3126328586747194693rm_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1185_rev__image__eqI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: nat,F3: produc357393685978478089rm_a_b > nat] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1186_rev__image__eqI,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,B2: term_a_b,F3: produc357393685978478089rm_a_b > term_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_2002304162058146776rm_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1187_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: produc357393685978478089rm_a_b,F3: nat > produc357393685978478089rm_a_b] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_7842151682427799154rm_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1188_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1189_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: term_a_b,F3: nat > term_a_b] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_nat_term_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1190_rev__image__eqI,axiom,
! [X: term_a_b,A: set_term_a_b,B2: produc357393685978478089rm_a_b,F3: term_a_b > produc357393685978478089rm_a_b] :
( ( member_term_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_170086233023854126rm_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1191_rev__image__eqI,axiom,
! [X: term_a_b,A: set_term_a_b,B2: nat,F3: term_a_b > nat] :
( ( member_term_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_term_a_b_nat @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1192_rev__image__eqI,axiom,
! [X: term_a_b,A: set_term_a_b,B2: term_a_b,F3: term_a_b > term_a_b] :
( ( member_term_a_b @ X @ A )
=> ( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_3079262707727038895rm_a_b @ F3 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1193_SUP__cong,axiom,
! [A: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b,C2: produc357393685978478089rm_a_b > nat,D3: produc357393685978478089rm_a_b > nat] :
( ( A = B6 )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ B6 )
=> ( ( C2 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ D3 @ B6 ) ) ) ) ) ).
% SUP_cong
thf(fact_1194_SUP__cong,axiom,
! [A: set_nat,B6: set_nat,C2: nat > nat,D3: nat > nat] :
( ( A = B6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B6 )
=> ( ( C2 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B6 ) ) ) ) ) ).
% SUP_cong
thf(fact_1195_SUP__cong,axiom,
! [A: set_term_a_b,B6: set_term_a_b,C2: term_a_b > nat,D3: term_a_b > nat] :
( ( A = B6 )
=> ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ B6 )
=> ( ( C2 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ D3 @ B6 ) ) ) ) ) ).
% SUP_cong
thf(fact_1196_range__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: term_a_b > produc357393685978478089rm_a_b,X: term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_170086233023854126rm_a_b @ F3 @ top_top_set_term_a_b ) ) ) ).
% range_eqI
thf(fact_1197_range__eqI,axiom,
! [B2: nat,F3: term_a_b > nat,X: term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_term_a_b_nat @ F3 @ top_top_set_term_a_b ) ) ) ).
% range_eqI
thf(fact_1198_range__eqI,axiom,
! [B2: term_a_b,F3: term_a_b > term_a_b,X: term_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_3079262707727038895rm_a_b @ F3 @ top_top_set_term_a_b ) ) ) ).
% range_eqI
thf(fact_1199_range__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: nat > produc357393685978478089rm_a_b,X: nat] :
( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_7842151682427799154rm_a_b @ F3 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1200_range__eqI,axiom,
! [B2: nat,F3: nat > nat,X: nat] :
( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1201_range__eqI,axiom,
! [B2: term_a_b,F3: nat > term_a_b,X: nat] :
( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_nat_term_a_b @ F3 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1202_range__eqI,axiom,
! [B2: produc357393685978478089rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member5869715511025134514rm_a_b @ B2 @ ( image_3126328586747194693rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) ) ) ).
% range_eqI
thf(fact_1203_range__eqI,axiom,
! [B2: nat,F3: produc357393685978478089rm_a_b > nat,X: produc357393685978478089rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member_nat @ B2 @ ( image_5783597275426814868_b_nat @ F3 @ top_to1314267278846557113rm_a_b ) ) ) ).
% range_eqI
thf(fact_1204_range__eqI,axiom,
! [B2: term_a_b,F3: produc357393685978478089rm_a_b > term_a_b,X: produc357393685978478089rm_a_b] :
( ( B2
= ( F3 @ X ) )
=> ( member_term_a_b @ B2 @ ( image_2002304162058146776rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) ) ) ).
% range_eqI
thf(fact_1205_rangeI,axiom,
! [F3: term_a_b > produc357393685978478089rm_a_b,X: term_a_b] : ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_170086233023854126rm_a_b @ F3 @ top_top_set_term_a_b ) ) ).
% rangeI
thf(fact_1206_rangeI,axiom,
! [F3: term_a_b > nat,X: term_a_b] : ( member_nat @ ( F3 @ X ) @ ( image_term_a_b_nat @ F3 @ top_top_set_term_a_b ) ) ).
% rangeI
thf(fact_1207_rangeI,axiom,
! [F3: term_a_b > term_a_b,X: term_a_b] : ( member_term_a_b @ ( F3 @ X ) @ ( image_3079262707727038895rm_a_b @ F3 @ top_top_set_term_a_b ) ) ).
% rangeI
thf(fact_1208_rangeI,axiom,
! [F3: nat > produc357393685978478089rm_a_b,X: nat] : ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_7842151682427799154rm_a_b @ F3 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1209_rangeI,axiom,
! [F3: nat > nat,X: nat] : ( member_nat @ ( F3 @ X ) @ ( image_nat_nat @ F3 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1210_rangeI,axiom,
! [F3: nat > term_a_b,X: nat] : ( member_term_a_b @ ( F3 @ X ) @ ( image_nat_term_a_b @ F3 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1211_rangeI,axiom,
! [F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,X: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ ( F3 @ X ) @ ( image_3126328586747194693rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) ) ).
% rangeI
thf(fact_1212_rangeI,axiom,
! [F3: produc357393685978478089rm_a_b > nat,X: produc357393685978478089rm_a_b] : ( member_nat @ ( F3 @ X ) @ ( image_5783597275426814868_b_nat @ F3 @ top_to1314267278846557113rm_a_b ) ) ).
% rangeI
thf(fact_1213_rangeI,axiom,
! [F3: produc357393685978478089rm_a_b > term_a_b,X: produc357393685978478089rm_a_b] : ( member_term_a_b @ ( F3 @ X ) @ ( image_2002304162058146776rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) ) ).
% rangeI
thf(fact_1214_Sup__SUP__eq,axiom,
( comple1058430234361567751_a_b_o
= ( ^ [S8: set_Pr2760896171883806314_a_b_o,X4: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X4 @ ( comple4970752389892389974rm_a_b @ ( image_5905906033996252654rm_a_b @ collec99177395925924084rm_a_b @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1215_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S8: set_nat_o,X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1216_Sup__SUP__eq,axiom,
( comple6164909459678505640_a_b_o
= ( ^ [S8: set_term_a_b_o,X4: term_a_b] : ( member_term_a_b @ X4 @ ( comple5064467279716934173rm_a_b @ ( image_8182850614055592384rm_a_b @ collect_term_a_b @ S8 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1217_subset__image__iff,axiom,
! [B6: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B6
= ( image_nat_nat @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1218_image__subset__iff,axiom,
! [F3: nat > nat,A: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B6 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F3 @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_1219_subset__imageE,axiom,
! [B6: set_nat,F3: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A )
=> ( B6
!= ( image_nat_nat @ F3 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_1220_image__subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le118470702582115849rm_a_b @ ( image_3126328586747194693rm_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1221_image__subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat,B6: set_nat] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_less_eq_set_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1222_image__subsetI,axiom,
! [A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > term_a_b,B6: set_term_a_b] :
( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( member_term_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( image_2002304162058146776rm_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1223_image__subsetI,axiom,
! [A: set_nat,F3: nat > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le118470702582115849rm_a_b @ ( image_7842151682427799154rm_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1224_image__subsetI,axiom,
! [A: set_nat,F3: nat > nat,B6: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1225_image__subsetI,axiom,
! [A: set_nat,F3: nat > term_a_b,B6: set_term_a_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_term_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( image_nat_term_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1226_image__subsetI,axiom,
! [A: set_term_a_b,F3: term_a_b > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( member5869715511025134514rm_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le118470702582115849rm_a_b @ ( image_170086233023854126rm_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1227_image__subsetI,axiom,
! [A: set_term_a_b,F3: term_a_b > nat,B6: set_nat] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( member_nat @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_less_eq_set_nat @ ( image_term_a_b_nat @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1228_image__subsetI,axiom,
! [A: set_term_a_b,F3: term_a_b > term_a_b,B6: set_term_a_b] :
( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( member_term_a_b @ ( F3 @ X2 ) @ B6 ) )
=> ( ord_le2705286416250468010rm_a_b @ ( image_3079262707727038895rm_a_b @ F3 @ A ) @ B6 ) ) ).
% image_subsetI
thf(fact_1229_image__mono,axiom,
! [A: set_nat,B6: set_nat,F3: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B6 ) ) ) ).
% image_mono
thf(fact_1230_image__Un,axiom,
! [F3: nat > nat,A: set_nat,B6: set_nat] :
( ( image_nat_nat @ F3 @ ( sup_sup_set_nat @ A @ B6 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B6 ) ) ) ).
% image_Un
thf(fact_1231_image__diff__subset,axiom,
! [F3: nat > nat,A: set_nat,B6: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B6 ) ) @ ( image_nat_nat @ F3 @ ( minus_minus_set_nat @ A @ B6 ) ) ) ).
% image_diff_subset
thf(fact_1232_image__Int__subset,axiom,
! [F3: nat > nat,A: set_nat,B6: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ ( inf_inf_set_nat @ A @ B6 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F3 @ A ) @ ( image_nat_nat @ F3 @ B6 ) ) ) ).
% image_Int_subset
thf(fact_1233_range__subsetD,axiom,
! [F3: term_a_b > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,I: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( image_170086233023854126rm_a_b @ F3 @ top_top_set_term_a_b ) @ B6 )
=> ( member5869715511025134514rm_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1234_range__subsetD,axiom,
! [F3: term_a_b > nat,B6: set_nat,I: term_a_b] :
( ( ord_less_eq_set_nat @ ( image_term_a_b_nat @ F3 @ top_top_set_term_a_b ) @ B6 )
=> ( member_nat @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1235_range__subsetD,axiom,
! [F3: term_a_b > term_a_b,B6: set_term_a_b,I: term_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( image_3079262707727038895rm_a_b @ F3 @ top_top_set_term_a_b ) @ B6 )
=> ( member_term_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1236_range__subsetD,axiom,
! [F3: nat > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,I: nat] :
( ( ord_le118470702582115849rm_a_b @ ( image_7842151682427799154rm_a_b @ F3 @ top_top_set_nat ) @ B6 )
=> ( member5869715511025134514rm_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1237_range__subsetD,axiom,
! [F3: nat > nat,B6: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ B6 )
=> ( member_nat @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1238_range__subsetD,axiom,
! [F3: nat > term_a_b,B6: set_term_a_b,I: nat] :
( ( ord_le2705286416250468010rm_a_b @ ( image_nat_term_a_b @ F3 @ top_top_set_nat ) @ B6 )
=> ( member_term_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1239_range__subsetD,axiom,
! [F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b,B6: set_Pr4386577575007340137rm_a_b,I: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( image_3126328586747194693rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) @ B6 )
=> ( member5869715511025134514rm_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1240_range__subsetD,axiom,
! [F3: produc357393685978478089rm_a_b > nat,B6: set_nat,I: produc357393685978478089rm_a_b] :
( ( ord_less_eq_set_nat @ ( image_5783597275426814868_b_nat @ F3 @ top_to1314267278846557113rm_a_b ) @ B6 )
=> ( member_nat @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1241_range__subsetD,axiom,
! [F3: produc357393685978478089rm_a_b > term_a_b,B6: set_term_a_b,I: produc357393685978478089rm_a_b] :
( ( ord_le2705286416250468010rm_a_b @ ( image_2002304162058146776rm_a_b @ F3 @ top_to1314267278846557113rm_a_b ) @ B6 )
=> ( member_term_a_b @ ( F3 @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_1242_the__elem__image__unique,axiom,
! [A: set_nat,F3: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F3 @ A ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1243_range__eq__singletonD,axiom,
! [F3: term_a_b > nat,A2: nat,X: term_a_b] :
( ( ( image_term_a_b_nat @ F3 @ top_top_set_term_a_b )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F3 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1244_range__eq__singletonD,axiom,
! [F3: nat > nat,A2: nat,X: nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F3 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1245_range__eq__singletonD,axiom,
! [F3: produc357393685978478089rm_a_b > nat,A2: nat,X: produc357393685978478089rm_a_b] :
( ( ( image_5783597275426814868_b_nat @ F3 @ top_to1314267278846557113rm_a_b )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F3 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1246_cSUP__least,axiom,
! [A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat,M3: nat] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1247_cSUP__least,axiom,
! [A: set_term_a_b,F3: term_a_b > nat,M3: nat] :
( ( A != bot_bot_set_term_a_b )
=> ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ F3 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1248_cSUP__least,axiom,
! [A: set_nat,F3: nat > nat,M3: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1249_in__image__insert__iff,axiom,
! [B6: set_se1467346398288745161rm_a_b,X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b] :
( ! [C6: set_Pr4386577575007340137rm_a_b] :
( ( member8823642877475723282rm_a_b @ C6 @ B6 )
=> ~ ( member5869715511025134514rm_a_b @ X @ C6 ) )
=> ( ( member8823642877475723282rm_a_b @ A @ ( image_8790183037202074949rm_a_b @ ( insert7009541432154983385rm_a_b @ X ) @ B6 ) )
= ( ( member5869715511025134514rm_a_b @ X @ A )
& ( member8823642877475723282rm_a_b @ ( minus_5192120951422937424rm_a_b @ A @ ( insert7009541432154983385rm_a_b @ X @ bot_bo197521221353338581rm_a_b ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1250_in__image__insert__iff,axiom,
! [B6: set_set_term_a_b,X: term_a_b,A: set_term_a_b] :
( ! [C6: set_term_a_b] :
( ( member_set_term_a_b @ C6 @ B6 )
=> ~ ( member_term_a_b @ X @ C6 ) )
=> ( ( member_set_term_a_b @ A @ ( image_7657200479700665115rm_a_b @ ( insert_term_a_b @ X ) @ B6 ) )
= ( ( member_term_a_b @ X @ A )
& ( member_set_term_a_b @ ( minus_4613853266328121507rm_a_b @ A @ ( insert_term_a_b @ X @ bot_bot_set_term_a_b ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1251_in__image__insert__iff,axiom,
! [B6: set_set_nat,X: nat,A: set_nat] :
( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ B6 )
=> ~ ( member_nat @ X @ C6 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B6 ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1252_cSUP__upper2,axiom,
! [F3: produc357393685978478089rm_a_b > nat,A: set_Pr4386577575007340137rm_a_b,X: produc357393685978478089rm_a_b,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) )
=> ( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F3 @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1253_cSUP__upper2,axiom,
! [F3: nat > nat,A: set_nat,X: nat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ( member_nat @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F3 @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1254_cSUP__upper2,axiom,
! [F3: term_a_b > nat,A: set_term_a_b,X: term_a_b,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_term_a_b_nat @ F3 @ A ) )
=> ( ( member_term_a_b @ X @ A )
=> ( ( ord_less_eq_nat @ U @ ( F3 @ X ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ F3 @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1255_cSUP__upper,axiom,
! [X: produc357393685978478089rm_a_b,A: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat] :
( ( member5869715511025134514rm_a_b @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1256_cSUP__upper,axiom,
! [X: nat,A: set_nat,F3: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1257_cSUP__upper,axiom,
! [X: term_a_b,A: set_term_a_b,F3: term_a_b > nat] :
( ( member_term_a_b @ X @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_term_a_b_nat @ F3 @ A ) )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ F3 @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1258_cSUP__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,G: nat > nat,B6: set_nat,F3: produc357393685978478089rm_a_b > nat] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B6 ) )
=> ( ! [N2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ N2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B6 )
& ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B6 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1259_cSUP__mono,axiom,
! [A: set_term_a_b,G: nat > nat,B6: set_nat,F3: term_a_b > nat] :
( ( A != bot_bot_set_term_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B6 ) )
=> ( ! [N2: term_a_b] :
( ( member_term_a_b @ N2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B6 )
& ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B6 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1260_cSUP__mono,axiom,
! [A: set_nat,G: nat > nat,B6: set_nat,F3: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B6 ) )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B6 )
& ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B6 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1261_cSUP__le__iff,axiom,
! [A: set_nat,F3: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F3 @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) @ U )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1262_cSUP__subset__mono,axiom,
! [A: set_Pr4386577575007340137rm_a_b,G: produc357393685978478089rm_a_b > nat,B6: set_Pr4386577575007340137rm_a_b,F3: produc357393685978478089rm_a_b > nat] :
( ( A != bot_bo197521221353338581rm_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_5783597275426814868_b_nat @ G @ B6 ) )
=> ( ( ord_le118470702582115849rm_a_b @ A @ B6 )
=> ( ! [X2: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_5783597275426814868_b_nat @ G @ B6 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1263_cSUP__subset__mono,axiom,
! [A: set_term_a_b,G: term_a_b > nat,B6: set_term_a_b,F3: term_a_b > nat] :
( ( A != bot_bot_set_term_a_b )
=> ( ( condit2214826472909112428ve_nat @ ( image_term_a_b_nat @ G @ B6 ) )
=> ( ( ord_le2705286416250468010rm_a_b @ A @ B6 )
=> ( ! [X2: term_a_b] :
( ( member_term_a_b @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_term_a_b_nat @ G @ B6 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1264_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > nat,B6: set_nat,F3: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B6 ) )
=> ( ( ord_less_eq_set_nat @ A @ B6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F3 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B6 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1265_surj__def,axiom,
! [F3: nat > nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: nat] :
( Y5
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1266_surj__def,axiom,
! [F3: nat > produc357393685978478089rm_a_b] :
( ( ( image_7842151682427799154rm_a_b @ F3 @ top_top_set_nat )
= top_to1314267278846557113rm_a_b )
= ( ! [Y5: produc357393685978478089rm_a_b] :
? [X4: nat] :
( Y5
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1267_surj__def,axiom,
! [F3: produc357393685978478089rm_a_b > term_a_b] :
( ( ( image_2002304162058146776rm_a_b @ F3 @ top_to1314267278846557113rm_a_b )
= top_top_set_term_a_b )
= ( ! [Y5: term_a_b] :
? [X4: produc357393685978478089rm_a_b] :
( Y5
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1268_surj__def,axiom,
! [F3: produc357393685978478089rm_a_b > nat] :
( ( ( image_5783597275426814868_b_nat @ F3 @ top_to1314267278846557113rm_a_b )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: produc357393685978478089rm_a_b] :
( Y5
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1269_surj__def,axiom,
! [F3: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b] :
( ( ( image_3126328586747194693rm_a_b @ F3 @ top_to1314267278846557113rm_a_b )
= top_to1314267278846557113rm_a_b )
= ( ! [Y5: produc357393685978478089rm_a_b] :
? [X4: produc357393685978478089rm_a_b] :
( Y5
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_1270_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1271_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1272_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1273_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1274_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
% Conjectures (1)
thf(conj_0,conjecture,
rewrit7370782028743737947rm_a_b @ ( sig_step_a_b @ f @ ( rstep_a_b @ r ) ) @ ( sig_step_a_b @ f @ ( rstep_a_b @ s ) ) @ top_top_set_term_a_b ).
%------------------------------------------------------------------------------