TPTP Problem File: ITP138^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP138^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_525__3267638_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Paraconsistency/prob_525__3267638_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 435 ( 153 unt; 80 typ; 0 def)
% Number of atoms : 994 ( 360 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 2958 ( 96 ~; 0 |; 84 &;2337 @)
% ( 0 <=>; 441 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 419 ( 419 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 68 usr; 10 con; 0-2 aty)
% Number of variables : 1164 ( 139 ^;1001 !; 24 ?;1164 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:45.032
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
produc696645085lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_se800614875lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__String__Ochar_J_J_J,type,
set_set_list_char: $tType ).
thf(ty_n_t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
set_Pa1764573435lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
set_list_char: $tType ).
thf(ty_n_t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
paraco1605129243lle_tv: $tType ).
thf(ty_n_t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
paraco1604210848lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (68)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__List__Olist_It__String__Ochar_J,type,
finite1803457147t_char: set_list_char > set_set_list_char ).
thf(sy_c_Finite__Set_OFpow_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
finite2011762305lle_tv: set_Pa1764573435lle_tv > set_se800614875lle_tv ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
ord_le1945897678char_o: ( list_char > $o ) > ( list_char > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_Eo_J,type,
ord_le1632188418e_tv_o: ( paraco1605129243lle_tv > $o ) > ( paraco1605129243lle_tv > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
ord_le1836582007t_char: set_list_char > set_list_char > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
ord_le241064091lle_tv: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le2085948664_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
ord_le263913851lle_tv: set_se800614875lle_tv > set_se800614875lle_tv > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_Eo_J,type,
order_Greatest_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
order_350719188lle_tv: ( set_Pa1764573435lle_tv > $o ) > set_Pa1764573435lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
top_top_list_char_o: list_char > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_Eo_J,type,
top_to1299250770e_tv_o: paraco1605129243lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
top_to748628679t_char: set_list_char ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
top_to1057771083lle_tv: set_Pa1764573435lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__String__Ochar_J_J_J,type,
top_to697846397t_char: set_set_list_char ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Odomain,type,
paraco1933902578domain: set_nat > set_Pa1764573435lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Oeval,type,
paraco1325918602e_eval: ( list_char > paraco1605129243lle_tv ) > paraco1604210848lle_fm > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OCon_H,type,
paraco69239110le_Con: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OEql,type,
paraco1389086619le_Eql: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OEql_H,type,
paraco1745535428le_Eql: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_ONeg_H,type,
paraco445776468le_Neg: paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OPro,type,
paraco1480028776le_Pro: list_char > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OTruth,type,
paraco367965286_Truth: paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Otv_ODet,type,
paraco1344940915le_Det: $o > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ovalid,type,
paraco327635598_valid: paraco1604210848lle_fm > $o ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ovalid__in,type,
paraco55203475lid_in: set_nat > paraco1604210848lle_fm > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
produc1710735575lle_fm: ( list_char > paraco1605129243lle_tv ) > paraco1604210848lle_fm > produc696645085lle_fm ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__String__Ochar_J,type,
collect_list_char: ( list_char > $o ) > set_list_char ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
collec1111687558lle_tv: ( paraco1605129243lle_tv > $o ) > set_Pa1764573435lle_tv ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001t__List__Olist_It__String__Ochar_J,type,
pow_list_char: set_list_char > set_set_list_char ).
thf(sy_c_Set_OPow_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
pow_Pa475453624lle_tv: set_Pa1764573435lle_tv > set_se800614875lle_tv ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__List__Olist_It__String__Ochar_J,type,
image_o_list_char: ( $o > list_char ) > set_o > set_list_char ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_1491822472lle_tv: ( $o > paraco1605129243lle_tv ) > set_o > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__String__Ochar_J_001_Eo,type,
image_list_char_o: ( list_char > $o ) > set_list_char > set_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
image_890964937t_char: ( list_char > list_char ) > set_list_char > set_list_char ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__String__Ochar_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_1187894195lle_tv: ( list_char > paraco1605129243lle_tv ) > set_list_char > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
image_nat_list_char: ( nat > list_char ) > set_nat > set_list_char ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_1151963140lle_tv: ( nat > paraco1605129243lle_tv ) > set_nat > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001_Eo,type,
image_991799618e_tv_o: ( paraco1605129243lle_tv > $o ) > set_Pa1764573435lle_tv > set_o ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__List__Olist_It__String__Ochar_J,type,
image_1971766583t_char: ( paraco1605129243lle_tv > list_char ) > set_Pa1764573435lle_tv > set_list_char ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
image_376325286tv_nat: ( paraco1605129243lle_tv > nat ) > set_Pa1764573435lle_tv > set_nat ).
thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_335818702lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > set_Pa1764573435lle_tv > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_30320477lle_tv: ( set_list_char > set_Pa1764573435lle_tv ) > set_set_list_char > set_se800614875lle_tv ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
member266900804lle_tv: paraco1605129243lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member1875402276lle_tv: set_Pa1764573435lle_tv > set_se800614875lle_tv > $o ).
thf(sy_v_p,type,
p: paraco1604210848lle_fm ).
% Relevant facts (354)
thf(fact_0_universal__domain,axiom,
( ( paraco1933902578domain
@ ( collect_nat
@ ^ [N: nat] : $true ) )
= ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] : $true ) ) ).
% universal_domain
thf(fact_1_tv_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( paraco1344940915le_Det @ X1 )
= ( paraco1344940915le_Det @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tv.inject(1)
thf(fact_2_valid__in__def,axiom,
( paraco55203475lid_in
= ( ^ [U: set_nat,P: paraco1604210848lle_fm] :
! [I: list_char > paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ I @ top_to748628679t_char ) @ ( paraco1933902578domain @ U ) )
=> ( ( paraco1325918602e_eval @ I @ P )
= ( paraco1344940915le_Det @ $true ) ) ) ) ) ).
% valid_in_def
thf(fact_3_image__ident,axiom,
! [Y: set_o] :
( ( image_o_o
@ ^ [X: $o] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_4_UNIV__I,axiom,
! [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ top_to1057771083lle_tv ) ).
% UNIV_I
thf(fact_5_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_6_UNIV__I,axiom,
! [X2: list_char] : ( member_list_char @ X2 @ top_to748628679t_char ) ).
% UNIV_I
thf(fact_7_iso__tuple__UNIV__I,axiom,
! [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ top_to1057771083lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_8_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_9_iso__tuple__UNIV__I,axiom,
! [X2: list_char] : ( member_list_char @ X2 @ top_to748628679t_char ) ).
% iso_tuple_UNIV_I
thf(fact_10_subsetI,axiom,
! [A: set_list_char,B: set_list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A )
=> ( member_list_char @ X3 @ B ) )
=> ( ord_le1836582007t_char @ A @ B ) ) ).
% subsetI
thf(fact_11_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_12_subsetI,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member266900804lle_tv @ X3 @ B ) )
=> ( ord_le241064091lle_tv @ A @ B ) ) ).
% subsetI
thf(fact_13_subset__antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_14_subset__antisym,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( ord_le241064091lle_tv @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_15_image__eqI,axiom,
! [B2: list_char,F: $o > list_char,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_list_char @ B2 @ ( image_o_list_char @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_16_image__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: $o > paraco1605129243lle_tv,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member266900804lle_tv @ B2 @ ( image_1491822472lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_17_image__eqI,axiom,
! [B2: $o,F: list_char > $o,X2: list_char,A: set_list_char] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_char @ X2 @ A )
=> ( member_o @ B2 @ ( image_list_char_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_18_image__eqI,axiom,
! [B2: list_char,F: list_char > list_char,X2: list_char,A: set_list_char] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_char @ X2 @ A )
=> ( member_list_char @ B2 @ ( image_890964937t_char @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_19_image__eqI,axiom,
! [B2: $o,F: paraco1605129243lle_tv > $o,X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( B2
= ( F @ X2 ) )
=> ( ( member266900804lle_tv @ X2 @ A )
=> ( member_o @ B2 @ ( image_991799618e_tv_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_20_image__eqI,axiom,
! [B2: list_char,F: paraco1605129243lle_tv > list_char,X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( B2
= ( F @ X2 ) )
=> ( ( member266900804lle_tv @ X2 @ A )
=> ( member_list_char @ B2 @ ( image_1971766583t_char @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_21_image__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( B2
= ( F @ X2 ) )
=> ( ( member266900804lle_tv @ X2 @ A )
=> ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_22_image__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv,X2: list_char,A: set_list_char] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_char @ X2 @ A )
=> ( member266900804lle_tv @ B2 @ ( image_1187894195lle_tv @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_23_image__eqI,axiom,
! [B2: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_24_order__refl,axiom,
! [X2: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ X2 @ X2 ) ).
% order_refl
thf(fact_25_order__refl,axiom,
! [X2: set_o] : ( ord_less_eq_set_o @ X2 @ X2 ) ).
% order_refl
thf(fact_26_rangeE,axiom,
! [B2: $o,F: list_char > $o] :
( ( member_o @ B2 @ ( image_list_char_o @ F @ top_to748628679t_char ) )
=> ~ ! [X3: list_char] :
( B2
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_27_rangeE,axiom,
! [B2: list_char,F: list_char > list_char] :
( ( member_list_char @ B2 @ ( image_890964937t_char @ F @ top_to748628679t_char ) )
=> ~ ! [X3: list_char] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_28_rangeE,axiom,
! [B2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ B2 @ ( image_1187894195lle_tv @ F @ top_to748628679t_char ) )
=> ~ ! [X3: list_char] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_29_rangeE,axiom,
! [B2: $o,F: $o > $o] :
( ( member_o @ B2 @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B2
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_30_rangeE,axiom,
! [B2: list_char,F: $o > list_char] :
( ( member_list_char @ B2 @ ( image_o_list_char @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_31_rangeE,axiom,
! [B2: paraco1605129243lle_tv,F: $o > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ B2 @ ( image_1491822472lle_tv @ F @ top_top_set_o ) )
=> ~ ! [X3: $o] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_32_range__composition,axiom,
! [F: $o > $o,G: list_char > $o] :
( ( image_list_char_o
@ ^ [X: list_char] : ( F @ ( G @ X ) )
@ top_to748628679t_char )
= ( image_o_o @ F @ ( image_list_char_o @ G @ top_to748628679t_char ) ) ) ).
% range_composition
thf(fact_33_range__composition,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: list_char > paraco1605129243lle_tv] :
( ( image_1187894195lle_tv
@ ^ [X: list_char] : ( F @ ( G @ X ) )
@ top_to748628679t_char )
= ( image_335818702lle_tv @ F @ ( image_1187894195lle_tv @ G @ top_to748628679t_char ) ) ) ).
% range_composition
thf(fact_34_range__composition,axiom,
! [F: list_char > paraco1605129243lle_tv,G: list_char > list_char] :
( ( image_1187894195lle_tv
@ ^ [X: list_char] : ( F @ ( G @ X ) )
@ top_to748628679t_char )
= ( image_1187894195lle_tv @ F @ ( image_890964937t_char @ G @ top_to748628679t_char ) ) ) ).
% range_composition
thf(fact_35_range__composition,axiom,
! [F: list_char > paraco1605129243lle_tv,G: $o > list_char] :
( ( image_1491822472lle_tv
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_1187894195lle_tv @ F @ ( image_o_list_char @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_36_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_37_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > list_char,B: set_list_char] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member_list_char @ ( F @ X3 ) @ B ) )
=> ( ord_le1836582007t_char @ ( image_nat_list_char @ F @ ( collect_nat @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_38_image__Collect__subsetI,axiom,
! [P2: paraco1605129243lle_tv > $o,F: paraco1605129243lle_tv > list_char,B: set_list_char] :
( ! [X3: paraco1605129243lle_tv] :
( ( P2 @ X3 )
=> ( member_list_char @ ( F @ X3 ) @ B ) )
=> ( ord_le1836582007t_char @ ( image_1971766583t_char @ F @ ( collec1111687558lle_tv @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_39_image__Collect__subsetI,axiom,
! [P2: list_char > $o,F: list_char > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: list_char] :
( ( P2 @ X3 )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ ( collect_list_char @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_40_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_1151963140lle_tv @ F @ ( collect_nat @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_41_image__Collect__subsetI,axiom,
! [P2: paraco1605129243lle_tv > $o,F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( P2 @ X3 )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_335818702lle_tv @ F @ ( collec1111687558lle_tv @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_42_image__Collect__subsetI,axiom,
! [P2: $o > $o,F: $o > $o,B: set_o] :
( ! [X3: $o] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ ( collect_o @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_43_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > $o,B: set_o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_44_image__Collect__subsetI,axiom,
! [P2: paraco1605129243lle_tv > $o,F: paraco1605129243lle_tv > $o,B: set_o] :
( ! [X3: paraco1605129243lle_tv] :
( ( P2 @ X3 )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_991799618e_tv_o @ F @ ( collec1111687558lle_tv @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_45_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_46_top__set__def,axiom,
( top_to1057771083lle_tv
= ( collec1111687558lle_tv @ top_to1299250770e_tv_o ) ) ).
% top_set_def
thf(fact_47_top__set__def,axiom,
( top_to748628679t_char
= ( collect_list_char @ top_top_list_char_o ) ) ).
% top_set_def
thf(fact_48_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_49_less__eq__set__def,axiom,
( ord_le1836582007t_char
= ( ^ [A2: set_list_char,B3: set_list_char] :
( ord_le1945897678char_o
@ ^ [X: list_char] : ( member_list_char @ X @ A2 )
@ ^ [X: list_char] : ( member_list_char @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_50_less__eq__set__def,axiom,
( ord_le241064091lle_tv
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
( ord_le1632188418e_tv_o
@ ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ A2 )
@ ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_51_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A2: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A2 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_52_dual__order_Oantisym,axiom,
! [B2: set_Pa1764573435lle_tv,A3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B2 @ A3 )
=> ( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_53_dual__order_Oantisym,axiom,
! [B2: set_o,A3: set_o] :
( ( ord_less_eq_set_o @ B2 @ A3 )
=> ( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_54_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Pa1764573435lle_tv,Z: set_Pa1764573435lle_tv] : Y2 = Z )
= ( ^ [A4: set_Pa1764573435lle_tv,B4: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B4 @ A4 )
& ( ord_le241064091lle_tv @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_55_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_o,Z: set_o] : Y2 = Z )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_56_dual__order_Otrans,axiom,
! [B2: set_Pa1764573435lle_tv,A3: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B2 @ A3 )
=> ( ( ord_le241064091lle_tv @ C @ B2 )
=> ( ord_le241064091lle_tv @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_57_dual__order_Otrans,axiom,
! [B2: set_o,A3: set_o,C: set_o] :
( ( ord_less_eq_set_o @ B2 @ A3 )
=> ( ( ord_less_eq_set_o @ C @ B2 )
=> ( ord_less_eq_set_o @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_58_dual__order_Orefl,axiom,
! [A3: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_60_order__trans,axiom,
! [X2: set_Pa1764573435lle_tv,Y3: set_Pa1764573435lle_tv,Z2: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X2 @ Y3 )
=> ( ( ord_le241064091lle_tv @ Y3 @ Z2 )
=> ( ord_le241064091lle_tv @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_61_order__trans,axiom,
! [X2: set_o,Y3: set_o,Z2: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y3 )
=> ( ( ord_less_eq_set_o @ Y3 @ Z2 )
=> ( ord_less_eq_set_o @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_62_order__class_Oorder_Oantisym,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ord_le241064091lle_tv @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_63_order__class_Oorder_Oantisym,axiom,
! [A3: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_64_ord__le__eq__trans,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_le241064091lle_tv @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_65_ord__le__eq__trans,axiom,
! [A3: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_o @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_66_ord__eq__le__trans,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A3 = B2 )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ord_le241064091lle_tv @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_67_ord__eq__le__trans,axiom,
! [A3: set_o,B2: set_o,C: set_o] :
( ( A3 = B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_68_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Pa1764573435lle_tv,Z: set_Pa1764573435lle_tv] : Y2 = Z )
= ( ^ [A4: set_Pa1764573435lle_tv,B4: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A4 @ B4 )
& ( ord_le241064091lle_tv @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_69_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_o,Z: set_o] : Y2 = Z )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
& ( ord_less_eq_set_o @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_70_antisym__conv,axiom,
! [Y3: set_Pa1764573435lle_tv,X2: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ Y3 @ X2 )
=> ( ( ord_le241064091lle_tv @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_71_antisym__conv,axiom,
! [Y3: set_o,X2: set_o] :
( ( ord_less_eq_set_o @ Y3 @ X2 )
=> ( ( ord_less_eq_set_o @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv
thf(fact_72_order_Otrans,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ord_le241064091lle_tv @ A3 @ C ) ) ) ).
% order.trans
thf(fact_73_order_Otrans,axiom,
! [A3: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ A3 @ C ) ) ) ).
% order.trans
thf(fact_74_eq__refl,axiom,
! [X2: set_Pa1764573435lle_tv,Y3: set_Pa1764573435lle_tv] :
( ( X2 = Y3 )
=> ( ord_le241064091lle_tv @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_75_eq__refl,axiom,
! [X2: set_o,Y3: set_o] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_o @ X2 @ Y3 ) ) ).
% eq_refl
thf(fact_76_antisym,axiom,
! [X2: set_Pa1764573435lle_tv,Y3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X2 @ Y3 )
=> ( ( ord_le241064091lle_tv @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_77_antisym,axiom,
! [X2: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y3 )
=> ( ( ord_less_eq_set_o @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% antisym
thf(fact_78_eq__iff,axiom,
( ( ^ [Y2: set_Pa1764573435lle_tv,Z: set_Pa1764573435lle_tv] : Y2 = Z )
= ( ^ [X: set_Pa1764573435lle_tv,Y4: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X @ Y4 )
& ( ord_le241064091lle_tv @ Y4 @ X ) ) ) ) ).
% eq_iff
thf(fact_79_eq__iff,axiom,
( ( ^ [Y2: set_o,Z: set_o] : Y2 = Z )
= ( ^ [X: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X @ Y4 )
& ( ord_less_eq_set_o @ Y4 @ X ) ) ) ) ).
% eq_iff
thf(fact_80_ord__le__eq__subst,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_o,C: set_o] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A3: set_o,B2: set_o,F: set_o > set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A3: set_o,B2: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A3: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A3: set_o,F: set_Pa1764573435lle_tv > set_o,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A3: set_Pa1764573435lle_tv,F: set_o > set_Pa1764573435lle_tv,B2: set_o,C: set_o] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A3: set_o,F: set_o > set_o,B2: set_o,C: set_o] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_order__subst2,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ord_le241064091lle_tv @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A3: set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_o,C: set_o] :
( ( ord_le241064091lle_tv @ A3 @ B2 )
=> ( ( ord_less_eq_set_o @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A3: set_o,B2: set_o,F: set_o > set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ord_le241064091lle_tv @ ( F @ B2 ) @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A3: set_o,B2: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
=> ( ( ord_less_eq_set_o @ ( F @ B2 ) @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst1,axiom,
! [A3: set_Pa1764573435lle_tv,F: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A3 @ ( F @ B2 ) )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_order__subst1,axiom,
! [A3: set_Pa1764573435lle_tv,F: set_o > set_Pa1764573435lle_tv,B2: set_o,C: set_o] :
( ( ord_le241064091lle_tv @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_le241064091lle_tv @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le241064091lle_tv @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A3: set_o,F: set_Pa1764573435lle_tv > set_o,B2: set_Pa1764573435lle_tv,C: set_Pa1764573435lle_tv] :
( ( ord_less_eq_set_o @ A3 @ ( F @ B2 ) )
=> ( ( ord_le241064091lle_tv @ B2 @ C )
=> ( ! [X3: set_Pa1764573435lle_tv,Y5: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A3: set_o,F: set_o > set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X3: set_o,Y5: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y5 )
=> ( ord_less_eq_set_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_o @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_97_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: list_char,F: $o > list_char] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_char @ B2 @ ( image_o_list_char @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_98_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: paraco1605129243lle_tv,F: $o > paraco1605129243lle_tv] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member266900804lle_tv @ B2 @ ( image_1491822472lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_99_rev__image__eqI,axiom,
! [X2: list_char,A: set_list_char,B2: $o,F: list_char > $o] :
( ( member_list_char @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_list_char_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_100_rev__image__eqI,axiom,
! [X2: list_char,A: set_list_char,B2: list_char,F: list_char > list_char] :
( ( member_list_char @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_char @ B2 @ ( image_890964937t_char @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_101_rev__image__eqI,axiom,
! [X2: list_char,A: set_list_char,B2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv] :
( ( member_list_char @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member266900804lle_tv @ B2 @ ( image_1187894195lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_102_rev__image__eqI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B2: $o,F: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_991799618e_tv_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_103_rev__image__eqI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B2: list_char,F: paraco1605129243lle_tv > list_char] :
( ( member266900804lle_tv @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_char @ B2 @ ( image_1971766583t_char @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_104_rev__image__eqI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_105_ball__imageD,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,P2: paraco1605129243lle_tv > $o] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ ( image_1187894195lle_tv @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X4: list_char] :
( ( member_list_char @ X4 @ A )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_106_ball__imageD,axiom,
! [F: $o > $o,A: set_o,P2: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_107_image__cong,axiom,
! [M: set_o,N2: set_o,F: $o > $o,G: $o > $o] :
( ( M = N2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_o_o @ F @ M )
= ( image_o_o @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_108_image__cong,axiom,
! [M: set_list_char,N2: set_list_char,F: list_char > paraco1605129243lle_tv,G: list_char > paraco1605129243lle_tv] :
( ( M = N2 )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1187894195lle_tv @ F @ M )
= ( image_1187894195lle_tv @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_109_mem__Collect__eq,axiom,
! [A3: $o,P2: $o > $o] :
( ( member_o @ A3 @ ( collect_o @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A3: list_char,P2: list_char > $o] :
( ( member_list_char @ A3 @ ( collect_list_char @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A3: paraco1605129243lle_tv,P2: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ A3 @ ( collec1111687558lle_tv @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A: set_list_char] :
( ( collect_list_char
@ ^ [X: list_char] : ( member_list_char @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A: set_Pa1764573435lle_tv] :
( ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_117_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_118_Collect__cong,axiom,
! [P2: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv > $o] :
( ! [X3: paraco1605129243lle_tv] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1111687558lle_tv @ P2 )
= ( collec1111687558lle_tv @ Q ) ) ) ).
% Collect_cong
thf(fact_119_bex__imageD,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,P2: paraco1605129243lle_tv > $o] :
( ? [X4: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X4 @ ( image_1187894195lle_tv @ F @ A ) )
& ( P2 @ X4 ) )
=> ? [X3: list_char] :
( ( member_list_char @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_120_bex__imageD,axiom,
! [F: $o > $o,A: set_o,P2: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A ) )
& ( P2 @ X4 ) )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_121_image__iff,axiom,
! [Z2: $o,F: $o > $o,A: set_o] :
( ( member_o @ Z2 @ ( image_o_o @ F @ A ) )
= ( ? [X: $o] :
( ( member_o @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_122_image__iff,axiom,
! [Z2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv,A: set_list_char] :
( ( member266900804lle_tv @ Z2 @ ( image_1187894195lle_tv @ F @ A ) )
= ( ? [X: list_char] :
( ( member_list_char @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_123_imageI,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_124_imageI,axiom,
! [X2: $o,A: set_o,F: $o > list_char] :
( ( member_o @ X2 @ A )
=> ( member_list_char @ ( F @ X2 ) @ ( image_o_list_char @ F @ A ) ) ) ).
% imageI
thf(fact_125_imageI,axiom,
! [X2: $o,A: set_o,F: $o > paraco1605129243lle_tv] :
( ( member_o @ X2 @ A )
=> ( member266900804lle_tv @ ( F @ X2 ) @ ( image_1491822472lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_126_imageI,axiom,
! [X2: list_char,A: set_list_char,F: list_char > $o] :
( ( member_list_char @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_list_char_o @ F @ A ) ) ) ).
% imageI
thf(fact_127_imageI,axiom,
! [X2: list_char,A: set_list_char,F: list_char > list_char] :
( ( member_list_char @ X2 @ A )
=> ( member_list_char @ ( F @ X2 ) @ ( image_890964937t_char @ F @ A ) ) ) ).
% imageI
thf(fact_128_imageI,axiom,
! [X2: list_char,A: set_list_char,F: list_char > paraco1605129243lle_tv] :
( ( member_list_char @ X2 @ A )
=> ( member266900804lle_tv @ ( F @ X2 ) @ ( image_1187894195lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_129_imageI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_991799618e_tv_o @ F @ A ) ) ) ).
% imageI
thf(fact_130_imageI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > list_char] :
( ( member266900804lle_tv @ X2 @ A )
=> ( member_list_char @ ( F @ X2 ) @ ( image_1971766583t_char @ F @ A ) ) ) ).
% imageI
thf(fact_131_imageI,axiom,
! [X2: paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X2 @ A )
=> ( member266900804lle_tv @ ( F @ X2 ) @ ( image_335818702lle_tv @ F @ A ) ) ) ).
% imageI
thf(fact_132_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_133_Collect__mono__iff,axiom,
! [P2: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv > $o] :
( ( ord_le241064091lle_tv @ ( collec1111687558lle_tv @ P2 ) @ ( collec1111687558lle_tv @ Q ) )
= ( ! [X: paraco1605129243lle_tv] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_134_Collect__mono__iff,axiom,
! [P2: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P2 ) @ ( collect_o @ Q ) )
= ( ! [X: $o] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_135_set__eq__subset,axiom,
( ( ^ [Y2: set_Pa1764573435lle_tv,Z: set_Pa1764573435lle_tv] : Y2 = Z )
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A2 @ B3 )
& ( ord_le241064091lle_tv @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_136_set__eq__subset,axiom,
( ( ^ [Y2: set_o,Z: set_o] : Y2 = Z )
= ( ^ [A2: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
& ( ord_less_eq_set_o @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_137_subset__trans,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,C2: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( ord_le241064091lle_tv @ B @ C2 )
=> ( ord_le241064091lle_tv @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_138_subset__trans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_139_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_140_Collect__mono,axiom,
! [P2: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv > $o] :
( ! [X3: paraco1605129243lle_tv] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le241064091lle_tv @ ( collec1111687558lle_tv @ P2 ) @ ( collec1111687558lle_tv @ Q ) ) ) ).
% Collect_mono
thf(fact_141_Collect__mono,axiom,
! [P2: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P2 ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_142_subset__refl,axiom,
! [A: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A @ A ) ).
% subset_refl
thf(fact_143_subset__refl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% subset_refl
thf(fact_144_subset__iff,axiom,
( ord_le1836582007t_char
= ( ^ [A2: set_list_char,B3: set_list_char] :
! [T: list_char] :
( ( member_list_char @ T @ A2 )
=> ( member_list_char @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_145_subset__iff,axiom,
( ord_le241064091lle_tv
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
! [T: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ T @ A2 )
=> ( member266900804lle_tv @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_146_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A2: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A2 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_147_equalityD2,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( A = B )
=> ( ord_le241064091lle_tv @ B @ A ) ) ).
% equalityD2
thf(fact_148_equalityD2,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ B @ A ) ) ).
% equalityD2
thf(fact_149_equalityD1,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( A = B )
=> ( ord_le241064091lle_tv @ A @ B ) ) ).
% equalityD1
thf(fact_150_equalityD1,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% equalityD1
thf(fact_151_subset__eq,axiom,
( ord_le1836582007t_char
= ( ^ [A2: set_list_char,B3: set_list_char] :
! [X: list_char] :
( ( member_list_char @ X @ A2 )
=> ( member_list_char @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_152_subset__eq,axiom,
( ord_le241064091lle_tv
= ( ^ [A2: set_Pa1764573435lle_tv,B3: set_Pa1764573435lle_tv] :
! [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A2 )
=> ( member266900804lle_tv @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_153_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A2: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_154_equalityE,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( A = B )
=> ~ ( ( ord_le241064091lle_tv @ A @ B )
=> ~ ( ord_le241064091lle_tv @ B @ A ) ) ) ).
% equalityE
thf(fact_155_equalityE,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_o @ A @ B )
=> ~ ( ord_less_eq_set_o @ B @ A ) ) ) ).
% equalityE
thf(fact_156_subsetD,axiom,
! [A: set_list_char,B: set_list_char,C: list_char] :
( ( ord_le1836582007t_char @ A @ B )
=> ( ( member_list_char @ C @ A )
=> ( member_list_char @ C @ B ) ) ) ).
% subsetD
thf(fact_157_subsetD,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,C: paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ C @ A )
=> ( member266900804lle_tv @ C @ B ) ) ) ).
% subsetD
thf(fact_158_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_159_in__mono,axiom,
! [A: set_list_char,B: set_list_char,X2: list_char] :
( ( ord_le1836582007t_char @ A @ B )
=> ( ( member_list_char @ X2 @ A )
=> ( member_list_char @ X2 @ B ) ) ) ).
% in_mono
thf(fact_160_in__mono,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,X2: paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ( member266900804lle_tv @ X2 @ A )
=> ( member266900804lle_tv @ X2 @ B ) ) ) ).
% in_mono
thf(fact_161_in__mono,axiom,
! [A: set_o,B: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B ) ) ) ).
% in_mono
thf(fact_162_UNIV__witness,axiom,
? [X3: paraco1605129243lle_tv] : ( member266900804lle_tv @ X3 @ top_to1057771083lle_tv ) ).
% UNIV_witness
thf(fact_163_UNIV__witness,axiom,
? [X3: list_char] : ( member_list_char @ X3 @ top_to748628679t_char ) ).
% UNIV_witness
thf(fact_164_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_165_UNIV__eq__I,axiom,
! [A: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] : ( member266900804lle_tv @ X3 @ A )
=> ( top_to1057771083lle_tv = A ) ) ).
% UNIV_eq_I
thf(fact_166_UNIV__eq__I,axiom,
! [A: set_list_char] :
( ! [X3: list_char] : ( member_list_char @ X3 @ A )
=> ( top_to748628679t_char = A ) ) ).
% UNIV_eq_I
thf(fact_167_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_168_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P2: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_169_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P2: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_170_Compr__image__eq,axiom,
! [F: paraco1605129243lle_tv > $o,A: set_Pa1764573435lle_tv,P2: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_991799618e_tv_o @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_991799618e_tv_o @ F
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_171_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P2: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_172_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_173_Compr__image__eq,axiom,
! [F: paraco1605129243lle_tv > nat,A: set_Pa1764573435lle_tv,P2: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_376325286tv_nat @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_376325286tv_nat @ F
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_174_Compr__image__eq,axiom,
! [F: $o > paraco1605129243lle_tv,A: set_o,P2: paraco1605129243lle_tv > $o] :
( ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ ( image_1491822472lle_tv @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_1491822472lle_tv @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_175_Compr__image__eq,axiom,
! [F: nat > paraco1605129243lle_tv,A: set_nat,P2: paraco1605129243lle_tv > $o] :
( ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ ( image_1151963140lle_tv @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_1151963140lle_tv @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_176_Compr__image__eq,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o] :
( ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ ( image_335818702lle_tv @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_335818702lle_tv @ F
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_177_Compr__image__eq,axiom,
! [F: list_char > $o,A: set_list_char,P2: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_list_char_o @ F @ A ) )
& ( P2 @ X ) ) )
= ( image_list_char_o @ F
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ A )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_178_image__image,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,G: list_char > paraco1605129243lle_tv,A: set_list_char] :
( ( image_335818702lle_tv @ F @ ( image_1187894195lle_tv @ G @ A ) )
= ( image_1187894195lle_tv
@ ^ [X: list_char] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_179_image__image,axiom,
! [F: list_char > paraco1605129243lle_tv,G: list_char > list_char,A: set_list_char] :
( ( image_1187894195lle_tv @ F @ ( image_890964937t_char @ G @ A ) )
= ( image_1187894195lle_tv
@ ^ [X: list_char] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_180_image__image,axiom,
! [F: $o > $o,G: $o > $o,A: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ A ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_181_imageE,axiom,
! [B2: $o,F: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_182_imageE,axiom,
! [B2: $o,F: list_char > $o,A: set_list_char] :
( ( member_o @ B2 @ ( image_list_char_o @ F @ A ) )
=> ~ ! [X3: list_char] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_list_char @ X3 @ A ) ) ) ).
% imageE
thf(fact_183_imageE,axiom,
! [B2: $o,F: paraco1605129243lle_tv > $o,A: set_Pa1764573435lle_tv] :
( ( member_o @ B2 @ ( image_991799618e_tv_o @ F @ A ) )
=> ~ ! [X3: paraco1605129243lle_tv] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member266900804lle_tv @ X3 @ A ) ) ) ).
% imageE
thf(fact_184_imageE,axiom,
! [B2: list_char,F: $o > list_char,A: set_o] :
( ( member_list_char @ B2 @ ( image_o_list_char @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_185_imageE,axiom,
! [B2: list_char,F: list_char > list_char,A: set_list_char] :
( ( member_list_char @ B2 @ ( image_890964937t_char @ F @ A ) )
=> ~ ! [X3: list_char] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_list_char @ X3 @ A ) ) ) ).
% imageE
thf(fact_186_imageE,axiom,
! [B2: list_char,F: paraco1605129243lle_tv > list_char,A: set_Pa1764573435lle_tv] :
( ( member_list_char @ B2 @ ( image_1971766583t_char @ F @ A ) )
=> ~ ! [X3: paraco1605129243lle_tv] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member266900804lle_tv @ X3 @ A ) ) ) ).
% imageE
thf(fact_187_imageE,axiom,
! [B2: paraco1605129243lle_tv,F: $o > paraco1605129243lle_tv,A: set_o] :
( ( member266900804lle_tv @ B2 @ ( image_1491822472lle_tv @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_188_imageE,axiom,
! [B2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv,A: set_list_char] :
( ( member266900804lle_tv @ B2 @ ( image_1187894195lle_tv @ F @ A ) )
=> ~ ! [X3: list_char] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_list_char @ X3 @ A ) ) ) ).
% imageE
thf(fact_189_imageE,axiom,
! [B2: paraco1605129243lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( member266900804lle_tv @ B2 @ ( image_335818702lle_tv @ F @ A ) )
=> ~ ! [X3: paraco1605129243lle_tv] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member266900804lle_tv @ X3 @ A ) ) ) ).
% imageE
thf(fact_190_Collect__restrict,axiom,
! [X5: set_list_char,P2: list_char > $o] :
( ord_le1836582007t_char
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_191_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_192_Collect__restrict,axiom,
! [X5: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o] :
( ord_le241064091lle_tv
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_193_Collect__restrict,axiom,
! [X5: set_o,P2: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_194_prop__restrict,axiom,
! [X2: list_char,Z3: set_list_char,X5: set_list_char,P2: list_char > $o] :
( ( member_list_char @ X2 @ Z3 )
=> ( ( ord_le1836582007t_char @ Z3
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_195_prop__restrict,axiom,
! [X2: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X2 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_196_prop__restrict,axiom,
! [X2: paraco1605129243lle_tv,Z3: set_Pa1764573435lle_tv,X5: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ X2 @ Z3 )
=> ( ( ord_le241064091lle_tv @ Z3
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_197_prop__restrict,axiom,
! [X2: $o,Z3: set_o,X5: set_o,P2: $o > $o] :
( ( member_o @ X2 @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_198_Collect__subset,axiom,
! [A: set_list_char,P2: list_char > $o] :
( ord_le1836582007t_char
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_199_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_200_Collect__subset,axiom,
! [A: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o] :
( ord_le241064091lle_tv
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_201_Collect__subset,axiom,
! [A: set_o,P2: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_202_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_203_UNIV__def,axiom,
( top_to1057771083lle_tv
= ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] : $true ) ) ).
% UNIV_def
thf(fact_204_UNIV__def,axiom,
( top_to748628679t_char
= ( collect_list_char
@ ^ [X: list_char] : $true ) ) ).
% UNIV_def
thf(fact_205_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_206_top_Oextremum__uniqueI,axiom,
! [A3: set_list_char] :
( ( ord_le1836582007t_char @ top_to748628679t_char @ A3 )
=> ( A3 = top_to748628679t_char ) ) ).
% top.extremum_uniqueI
thf(fact_207_top_Oextremum__uniqueI,axiom,
! [A3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A3 )
=> ( A3 = top_to1057771083lle_tv ) ) ).
% top.extremum_uniqueI
thf(fact_208_top_Oextremum__uniqueI,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A3 )
=> ( A3 = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_209_top_Oextremum__unique,axiom,
! [A3: set_list_char] :
( ( ord_le1836582007t_char @ top_to748628679t_char @ A3 )
= ( A3 = top_to748628679t_char ) ) ).
% top.extremum_unique
thf(fact_210_top_Oextremum__unique,axiom,
! [A3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A3 )
= ( A3 = top_to1057771083lle_tv ) ) ).
% top.extremum_unique
thf(fact_211_top_Oextremum__unique,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A3 )
= ( A3 = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_212_top__greatest,axiom,
! [A3: set_list_char] : ( ord_le1836582007t_char @ A3 @ top_to748628679t_char ) ).
% top_greatest
thf(fact_213_top__greatest,axiom,
! [A3: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A3 @ top_to1057771083lle_tv ) ).
% top_greatest
thf(fact_214_top__greatest,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ top_top_set_o ) ).
% top_greatest
thf(fact_215_subset__image__iff,axiom,
! [B: set_Pa1764573435lle_tv,F: list_char > paraco1605129243lle_tv,A: set_list_char] :
( ( ord_le241064091lle_tv @ B @ ( image_1187894195lle_tv @ F @ A ) )
= ( ? [AA: set_list_char] :
( ( ord_le1836582007t_char @ AA @ A )
& ( B
= ( image_1187894195lle_tv @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_216_subset__image__iff,axiom,
! [B: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B @ ( image_335818702lle_tv @ F @ A ) )
= ( ? [AA: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ AA @ A )
& ( B
= ( image_335818702lle_tv @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_217_subset__image__iff,axiom,
! [B: set_Pa1764573435lle_tv,F: $o > paraco1605129243lle_tv,A: set_o] :
( ( ord_le241064091lle_tv @ B @ ( image_1491822472lle_tv @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_1491822472lle_tv @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_218_subset__image__iff,axiom,
! [B: set_o,F: paraco1605129243lle_tv > $o,A: set_Pa1764573435lle_tv] :
( ( ord_less_eq_set_o @ B @ ( image_991799618e_tv_o @ F @ A ) )
= ( ? [AA: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ AA @ A )
& ( B
= ( image_991799618e_tv_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_219_subset__image__iff,axiom,
! [B: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_220_image__subset__iff,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ A ) @ B )
= ( ! [X: list_char] :
( ( member_list_char @ X @ A )
=> ( member266900804lle_tv @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_221_image__subset__iff,axiom,
! [F: $o > $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_222_subset__imageE,axiom,
! [B: set_Pa1764573435lle_tv,F: list_char > paraco1605129243lle_tv,A: set_list_char] :
( ( ord_le241064091lle_tv @ B @ ( image_1187894195lle_tv @ F @ A ) )
=> ~ ! [C3: set_list_char] :
( ( ord_le1836582007t_char @ C3 @ A )
=> ( B
!= ( image_1187894195lle_tv @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_223_subset__imageE,axiom,
! [B: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B @ ( image_335818702lle_tv @ F @ A ) )
=> ~ ! [C3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ C3 @ A )
=> ( B
!= ( image_335818702lle_tv @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_224_subset__imageE,axiom,
! [B: set_Pa1764573435lle_tv,F: $o > paraco1605129243lle_tv,A: set_o] :
( ( ord_le241064091lle_tv @ B @ ( image_1491822472lle_tv @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_1491822472lle_tv @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_225_subset__imageE,axiom,
! [B: set_o,F: paraco1605129243lle_tv > $o,A: set_Pa1764573435lle_tv] :
( ( ord_less_eq_set_o @ B @ ( image_991799618e_tv_o @ F @ A ) )
=> ~ ! [C3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ C3 @ A )
=> ( B
!= ( image_991799618e_tv_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_226_subset__imageE,axiom,
! [B: set_o,F: $o > $o,A: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_227_image__subsetI,axiom,
! [A: set_o,F: $o > list_char,B: set_list_char] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_list_char @ ( F @ X3 ) @ B ) )
=> ( ord_le1836582007t_char @ ( image_o_list_char @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_228_image__subsetI,axiom,
! [A: set_list_char,F: list_char > list_char,B: set_list_char] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A )
=> ( member_list_char @ ( F @ X3 ) @ B ) )
=> ( ord_le1836582007t_char @ ( image_890964937t_char @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_229_image__subsetI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > list_char,B: set_list_char] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member_list_char @ ( F @ X3 ) @ B ) )
=> ( ord_le1836582007t_char @ ( image_1971766583t_char @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_230_image__subsetI,axiom,
! [A: set_o,F: $o > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_1491822472lle_tv @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_231_image__subsetI,axiom,
! [A: set_list_char,F: list_char > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_232_image__subsetI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv,B: set_Pa1764573435lle_tv] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member266900804lle_tv @ ( F @ X3 ) @ B ) )
=> ( ord_le241064091lle_tv @ ( image_335818702lle_tv @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_233_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_234_image__subsetI,axiom,
! [A: set_list_char,F: list_char > $o,B: set_o] :
( ! [X3: list_char] :
( ( member_list_char @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_list_char_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_235_image__subsetI,axiom,
! [A: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > $o,B: set_o] :
( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_991799618e_tv_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_236_image__mono,axiom,
! [A: set_list_char,B: set_list_char,F: list_char > paraco1605129243lle_tv] :
( ( ord_le1836582007t_char @ A @ B )
=> ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ A ) @ ( image_1187894195lle_tv @ F @ B ) ) ) ).
% image_mono
thf(fact_237_image__mono,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ord_le241064091lle_tv @ ( image_335818702lle_tv @ F @ A ) @ ( image_335818702lle_tv @ F @ B ) ) ) ).
% image_mono
thf(fact_238_image__mono,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv,F: paraco1605129243lle_tv > $o] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ord_less_eq_set_o @ ( image_991799618e_tv_o @ F @ A ) @ ( image_991799618e_tv_o @ F @ B ) ) ) ).
% image_mono
thf(fact_239_image__mono,axiom,
! [A: set_o,B: set_o,F: $o > paraco1605129243lle_tv] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_le241064091lle_tv @ ( image_1491822472lle_tv @ F @ A ) @ ( image_1491822472lle_tv @ F @ B ) ) ) ).
% image_mono
thf(fact_240_image__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ ( image_o_o @ F @ B ) ) ) ).
% image_mono
thf(fact_241_range__eqI,axiom,
! [B2: $o,F: list_char > $o,X2: list_char] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_list_char_o @ F @ top_to748628679t_char ) ) ) ).
% range_eqI
thf(fact_242_range__eqI,axiom,
! [B2: list_char,F: list_char > list_char,X2: list_char] :
( ( B2
= ( F @ X2 ) )
=> ( member_list_char @ B2 @ ( image_890964937t_char @ F @ top_to748628679t_char ) ) ) ).
% range_eqI
thf(fact_243_range__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: list_char > paraco1605129243lle_tv,X2: list_char] :
( ( B2
= ( F @ X2 ) )
=> ( member266900804lle_tv @ B2 @ ( image_1187894195lle_tv @ F @ top_to748628679t_char ) ) ) ).
% range_eqI
thf(fact_244_range__eqI,axiom,
! [B2: $o,F: $o > $o,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_245_range__eqI,axiom,
! [B2: list_char,F: $o > list_char,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member_list_char @ B2 @ ( image_o_list_char @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_246_range__eqI,axiom,
! [B2: paraco1605129243lle_tv,F: $o > paraco1605129243lle_tv,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member266900804lle_tv @ B2 @ ( image_1491822472lle_tv @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_247_rangeI,axiom,
! [F: list_char > $o,X2: list_char] : ( member_o @ ( F @ X2 ) @ ( image_list_char_o @ F @ top_to748628679t_char ) ) ).
% rangeI
thf(fact_248_rangeI,axiom,
! [F: list_char > list_char,X2: list_char] : ( member_list_char @ ( F @ X2 ) @ ( image_890964937t_char @ F @ top_to748628679t_char ) ) ).
% rangeI
thf(fact_249_rangeI,axiom,
! [F: list_char > paraco1605129243lle_tv,X2: list_char] : ( member266900804lle_tv @ ( F @ X2 ) @ ( image_1187894195lle_tv @ F @ top_to748628679t_char ) ) ).
% rangeI
thf(fact_250_rangeI,axiom,
! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_251_rangeI,axiom,
! [F: $o > list_char,X2: $o] : ( member_list_char @ ( F @ X2 ) @ ( image_o_list_char @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_252_rangeI,axiom,
! [F: $o > paraco1605129243lle_tv,X2: $o] : ( member266900804lle_tv @ ( F @ X2 ) @ ( image_1491822472lle_tv @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_253_subset__UNIV,axiom,
! [A: set_list_char] : ( ord_le1836582007t_char @ A @ top_to748628679t_char ) ).
% subset_UNIV
thf(fact_254_subset__UNIV,axiom,
! [A: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A @ top_to1057771083lle_tv ) ).
% subset_UNIV
thf(fact_255_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_256_surj__def,axiom,
! [F: list_char > paraco1605129243lle_tv] :
( ( ( image_1187894195lle_tv @ F @ top_to748628679t_char )
= top_to1057771083lle_tv )
= ( ! [Y4: paraco1605129243lle_tv] :
? [X: list_char] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_257_surj__def,axiom,
! [F: list_char > list_char] :
( ( ( image_890964937t_char @ F @ top_to748628679t_char )
= top_to748628679t_char )
= ( ! [Y4: list_char] :
? [X: list_char] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_258_surj__def,axiom,
! [F: list_char > $o] :
( ( ( image_list_char_o @ F @ top_to748628679t_char )
= top_top_set_o )
= ( ! [Y4: $o] :
? [X: list_char] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_259_surj__def,axiom,
! [F: $o > list_char] :
( ( ( image_o_list_char @ F @ top_top_set_o )
= top_to748628679t_char )
= ( ! [Y4: list_char] :
? [X: $o] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_260_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y4: $o] :
? [X: $o] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_261_surjI,axiom,
! [G: list_char > paraco1605129243lle_tv,F: paraco1605129243lle_tv > list_char] :
( ! [X3: paraco1605129243lle_tv] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_1187894195lle_tv @ G @ top_to748628679t_char )
= top_to1057771083lle_tv ) ) ).
% surjI
thf(fact_262_surjI,axiom,
! [G: list_char > list_char,F: list_char > list_char] :
( ! [X3: list_char] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_890964937t_char @ G @ top_to748628679t_char )
= top_to748628679t_char ) ) ).
% surjI
thf(fact_263_surjI,axiom,
! [G: list_char > $o,F: $o > list_char] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_list_char_o @ G @ top_to748628679t_char )
= top_top_set_o ) ) ).
% surjI
thf(fact_264_surjI,axiom,
! [G: $o > list_char,F: list_char > $o] :
( ! [X3: list_char] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_list_char @ G @ top_top_set_o )
= top_to748628679t_char ) ) ).
% surjI
thf(fact_265_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_266_surjE,axiom,
! [F: list_char > paraco1605129243lle_tv,Y3: paraco1605129243lle_tv] :
( ( ( image_1187894195lle_tv @ F @ top_to748628679t_char )
= top_to1057771083lle_tv )
=> ~ ! [X3: list_char] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_267_surjE,axiom,
! [F: list_char > list_char,Y3: list_char] :
( ( ( image_890964937t_char @ F @ top_to748628679t_char )
= top_to748628679t_char )
=> ~ ! [X3: list_char] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_268_surjE,axiom,
! [F: list_char > $o,Y3: $o] :
( ( ( image_list_char_o @ F @ top_to748628679t_char )
= top_top_set_o )
=> ~ ! [X3: list_char] :
( Y3
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_269_surjE,axiom,
! [F: $o > list_char,Y3: list_char] :
( ( ( image_o_list_char @ F @ top_top_set_o )
= top_to748628679t_char )
=> ~ ! [X3: $o] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_270_surjE,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X3: $o] :
( Y3
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_271_surjD,axiom,
! [F: list_char > paraco1605129243lle_tv,Y3: paraco1605129243lle_tv] :
( ( ( image_1187894195lle_tv @ F @ top_to748628679t_char )
= top_to1057771083lle_tv )
=> ? [X3: list_char] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_272_surjD,axiom,
! [F: list_char > list_char,Y3: list_char] :
( ( ( image_890964937t_char @ F @ top_to748628679t_char )
= top_to748628679t_char )
=> ? [X3: list_char] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_273_surjD,axiom,
! [F: list_char > $o,Y3: $o] :
( ( ( image_list_char_o @ F @ top_to748628679t_char )
= top_top_set_o )
=> ? [X3: list_char] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_274_surjD,axiom,
! [F: $o > list_char,Y3: list_char] :
( ( ( image_o_list_char @ F @ top_top_set_o )
= top_to748628679t_char )
=> ? [X3: $o] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_275_surjD,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X3: $o] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_276_all__subset__image,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,P2: set_Pa1764573435lle_tv > $o] :
( ( ! [B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B3 @ ( image_1187894195lle_tv @ F @ A ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_list_char] :
( ( ord_le1836582007t_char @ B3 @ A )
=> ( P2 @ ( image_1187894195lle_tv @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_277_all__subset__image,axiom,
! [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A: set_Pa1764573435lle_tv,P2: set_Pa1764573435lle_tv > $o] :
( ( ! [B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B3 @ ( image_335818702lle_tv @ F @ A ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B3 @ A )
=> ( P2 @ ( image_335818702lle_tv @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_278_all__subset__image,axiom,
! [F: $o > paraco1605129243lle_tv,A: set_o,P2: set_Pa1764573435lle_tv > $o] :
( ( ! [B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B3 @ ( image_1491822472lle_tv @ F @ A ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A )
=> ( P2 @ ( image_1491822472lle_tv @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_279_all__subset__image,axiom,
! [F: paraco1605129243lle_tv > $o,A: set_Pa1764573435lle_tv,P2: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_991799618e_tv_o @ F @ A ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ B3 @ A )
=> ( P2 @ ( image_991799618e_tv_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_280_all__subset__image,axiom,
! [F: $o > $o,A: set_o,P2: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_o_o @ F @ A ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A )
=> ( P2 @ ( image_o_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_281_valid__def,axiom,
( paraco327635598_valid
= ( ^ [P: paraco1604210848lle_fm] :
! [I: list_char > paraco1605129243lle_tv] :
( ( paraco1325918602e_eval @ I @ P )
= ( paraco1344940915le_Det @ $true ) ) ) ) ).
% valid_def
thf(fact_282_pred__subset__eq,axiom,
! [R: set_list_char,S: set_list_char] :
( ( ord_le1945897678char_o
@ ^ [X: list_char] : ( member_list_char @ X @ R )
@ ^ [X: list_char] : ( member_list_char @ X @ S ) )
= ( ord_le1836582007t_char @ R @ S ) ) ).
% pred_subset_eq
thf(fact_283_pred__subset__eq,axiom,
! [R: set_Pa1764573435lle_tv,S: set_Pa1764573435lle_tv] :
( ( ord_le1632188418e_tv_o
@ ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ R )
@ ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ S ) )
= ( ord_le241064091lle_tv @ R @ S ) ) ).
% pred_subset_eq
thf(fact_284_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ R )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_285_subset__Collect__iff,axiom,
! [B: set_list_char,A: set_list_char,P2: list_char > $o] :
( ( ord_le1836582007t_char @ B @ A )
=> ( ( ord_le1836582007t_char @ B
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: list_char] :
( ( member_list_char @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_286_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_287_subset__Collect__iff,axiom,
! [B: set_Pa1764573435lle_tv,A: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o] :
( ( ord_le241064091lle_tv @ B @ A )
=> ( ( ord_le241064091lle_tv @ B
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_288_subset__Collect__iff,axiom,
! [B: set_o,A: set_o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_289_subset__CollectI,axiom,
! [B: set_list_char,A: set_list_char,Q: list_char > $o,P2: list_char > $o] :
( ( ord_le1836582007t_char @ B @ A )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le1836582007t_char
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ B )
& ( Q @ X ) ) )
@ ( collect_list_char
@ ^ [X: list_char] :
( ( member_list_char @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_290_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_291_subset__CollectI,axiom,
! [B: set_Pa1764573435lle_tv,A: set_Pa1764573435lle_tv,Q: paraco1605129243lle_tv > $o,P2: paraco1605129243lle_tv > $o] :
( ( ord_le241064091lle_tv @ B @ A )
=> ( ! [X3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le241064091lle_tv
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ B )
& ( Q @ X ) ) )
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_292_subset__CollectI,axiom,
! [B: set_o,A: set_o,Q: $o > $o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ B )
& ( Q @ X ) ) )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_293_conj__subset__def,axiom,
! [A: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_294_conj__subset__def,axiom,
! [A: set_Pa1764573435lle_tv,P2: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv > $o] :
( ( ord_le241064091lle_tv @ A
@ ( collec1111687558lle_tv
@ ^ [X: paraco1605129243lle_tv] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le241064091lle_tv @ A @ ( collec1111687558lle_tv @ P2 ) )
& ( ord_le241064091lle_tv @ A @ ( collec1111687558lle_tv @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_295_conj__subset__def,axiom,
! [A: set_o,P2: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A
@ ( collect_o
@ ^ [X: $o] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_o @ A @ ( collect_o @ P2 ) )
& ( ord_less_eq_set_o @ A @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_296_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A: set_o] :
( ( Sup
@ ( image_o_o
@ ^ [X: $o] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_297_Inf_OINF__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A = B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_o_o @ C2 @ A ) )
= ( Inf @ ( image_o_o @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_298_Inf_OINF__cong,axiom,
! [A: set_list_char,B: set_list_char,C2: list_char > paraco1605129243lle_tv,D: list_char > paraco1605129243lle_tv,Inf: set_Pa1764573435lle_tv > paraco1605129243lle_tv] :
( ( A = B )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_1187894195lle_tv @ C2 @ A ) )
= ( Inf @ ( image_1187894195lle_tv @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_299_Sup_OSUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A = B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_o_o @ C2 @ A ) )
= ( Sup @ ( image_o_o @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_300_Sup_OSUP__cong,axiom,
! [A: set_list_char,B: set_list_char,C2: list_char > paraco1605129243lle_tv,D: list_char > paraco1605129243lle_tv,Sup: set_Pa1764573435lle_tv > paraco1605129243lle_tv] :
( ( A = B )
=> ( ! [X3: list_char] :
( ( member_list_char @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_1187894195lle_tv @ C2 @ A ) )
= ( Sup @ ( image_1187894195lle_tv @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_301_top__empty__eq,axiom,
( top_to1299250770e_tv_o
= ( ^ [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ top_to1057771083lle_tv ) ) ) ).
% top_empty_eq
thf(fact_302_top__empty__eq,axiom,
( top_top_list_char_o
= ( ^ [X: list_char] : ( member_list_char @ X @ top_to748628679t_char ) ) ) ).
% top_empty_eq
thf(fact_303_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_304_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A: set_o] :
( ( Inf
@ ( image_o_o
@ ^ [X: $o] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_305_eval_Osimps_I2_J,axiom,
! [I2: list_char > paraco1605129243lle_tv] :
( ( paraco1325918602e_eval @ I2 @ paraco367965286_Truth )
= ( paraco1344940915le_Det @ $true ) ) ).
% eval.simps(2)
thf(fact_306_image__Fpow__mono,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ A ) @ B )
=> ( ord_le263913851lle_tv @ ( image_30320477lle_tv @ ( image_1187894195lle_tv @ F ) @ ( finite1803457147t_char @ A ) ) @ ( finite2011762305lle_tv @ B ) ) ) ).
% image_Fpow_mono
thf(fact_307_image__Fpow__mono,axiom,
! [F: $o > $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B )
=> ( ord_le2085948664_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ ( finite_Fpow_o @ A ) ) @ ( finite_Fpow_o @ B ) ) ) ).
% image_Fpow_mono
thf(fact_308_eval_Osimps_I5_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,P3: paraco1604210848lle_fm,Q2: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P3 @ Q2 ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P3 @ Q2 ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ).
% eval.simps(5)
thf(fact_309_fm_Oinject_I4_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm,Y51: paraco1604210848lle_fm,Y52: paraco1604210848lle_fm] :
( ( ( paraco1389086619le_Eql @ X51 @ X52 )
= ( paraco1389086619le_Eql @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_310_fm_Odistinct_I15_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_311_Fpow__mono,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( ord_le263913851lle_tv @ ( finite2011762305lle_tv @ A ) @ ( finite2011762305lle_tv @ B ) ) ) ).
% Fpow_mono
thf(fact_312_Fpow__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_le2085948664_set_o @ ( finite_Fpow_o @ A ) @ ( finite_Fpow_o @ B ) ) ) ).
% Fpow_mono
thf(fact_313_eval__equality__simplify,axiom,
! [I2: list_char > paraco1605129243lle_tv,P3: paraco1604210848lle_fm,Q2: paraco1604210848lle_fm] :
( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P3 @ Q2 ) )
= ( paraco1344940915le_Det
@ ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1325918602e_eval @ I2 @ Q2 ) ) ) ) ).
% eval_equality_simplify
thf(fact_314_image__Pow__mono,axiom,
! [F: list_char > paraco1605129243lle_tv,A: set_list_char,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ F @ A ) @ B )
=> ( ord_le263913851lle_tv @ ( image_30320477lle_tv @ ( image_1187894195lle_tv @ F ) @ ( pow_list_char @ A ) ) @ ( pow_Pa475453624lle_tv @ B ) ) ) ).
% image_Pow_mono
thf(fact_315_image__Pow__mono,axiom,
! [F: $o > $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B )
=> ( ord_le2085948664_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ ( pow_o @ A ) ) @ ( pow_o @ B ) ) ) ).
% image_Pow_mono
thf(fact_316_eval_Osimps_I4_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,P3: paraco1604210848lle_fm,Q2: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ Q2 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P3 @ Q2 ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ) ) ) ) ).
% eval.simps(4)
thf(fact_317_Greatest__equality,axiom,
! [P2: set_Pa1764573435lle_tv > $o,X2: set_Pa1764573435lle_tv] :
( ( P2 @ X2 )
=> ( ! [Y5: set_Pa1764573435lle_tv] :
( ( P2 @ Y5 )
=> ( ord_le241064091lle_tv @ Y5 @ X2 ) )
=> ( ( order_350719188lle_tv @ P2 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_318_Greatest__equality,axiom,
! [P2: set_o > $o,X2: set_o] :
( ( P2 @ X2 )
=> ( ! [Y5: set_o] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_o @ Y5 @ X2 ) )
=> ( ( order_Greatest_set_o @ P2 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_319_GreatestI2__order,axiom,
! [P2: set_Pa1764573435lle_tv > $o,X2: set_Pa1764573435lle_tv,Q: set_Pa1764573435lle_tv > $o] :
( ( P2 @ X2 )
=> ( ! [Y5: set_Pa1764573435lle_tv] :
( ( P2 @ Y5 )
=> ( ord_le241064091lle_tv @ Y5 @ X2 ) )
=> ( ! [X3: set_Pa1764573435lle_tv] :
( ( P2 @ X3 )
=> ( ! [Y6: set_Pa1764573435lle_tv] :
( ( P2 @ Y6 )
=> ( ord_le241064091lle_tv @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_350719188lle_tv @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_320_GreatestI2__order,axiom,
! [P2: set_o > $o,X2: set_o,Q: set_o > $o] :
( ( P2 @ X2 )
=> ( ! [Y5: set_o] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_o @ Y5 @ X2 ) )
=> ( ! [X3: set_o] :
( ( P2 @ X3 )
=> ( ! [Y6: set_o] :
( ( P2 @ Y6 )
=> ( ord_less_eq_set_o @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_set_o @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_321_fm_Oinject_I3_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,Y41: paraco1604210848lle_fm,Y42: paraco1604210848lle_fm] :
( ( ( paraco69239110le_Con @ X41 @ X42 )
= ( paraco69239110le_Con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_322_PowI,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ A @ B )
=> ( member1875402276lle_tv @ A @ ( pow_Pa475453624lle_tv @ B ) ) ) ).
% PowI
thf(fact_323_PowI,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( member_set_o @ A @ ( pow_o @ B ) ) ) ).
% PowI
thf(fact_324_Pow__iff,axiom,
! [A: set_Pa1764573435lle_tv,B: set_Pa1764573435lle_tv] :
( ( member1875402276lle_tv @ A @ ( pow_Pa475453624lle_tv @ B ) )
= ( ord_le241064091lle_tv @ A @ B ) ) ).
% Pow_iff
thf(fact_325_Pow__iff,axiom,
! [A: set_o,B: set_o] :
( ( member_set_o @ A @ ( pow_o @ B ) )
= ( ord_less_eq_set_o @ A @ B ) ) ).
% Pow_iff
thf(fact_326_Pow__UNIV,axiom,
( ( pow_list_char @ top_to748628679t_char )
= top_to697846397t_char ) ).
% Pow_UNIV
thf(fact_327_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_328_fm_Odistinct_I25_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco69239110le_Con @ X41 @ X42 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_329_fm_Odistinct_I13_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_330_conjunction,axiom,
! [P3: paraco1604210848lle_fm,Q2: paraco1604210848lle_fm] :
( ( paraco327635598_valid @ ( paraco69239110le_Con @ P3 @ Q2 ) )
= ( ( paraco327635598_valid @ P3 )
& ( paraco327635598_valid @ Q2 ) ) ) ).
% conjunction
thf(fact_331_eval__negation,axiom,
! [I2: list_char > paraco1605129243lle_tv,P3: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P3 ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P3 ) )
= ( paraco1344940915le_Det @ $false ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P3 ) )
= ( paraco1325918602e_eval @ I2 @ P3 ) ) ) ) ) ) ).
% eval_negation
thf(fact_332_fm_Oinject_I2_J,axiom,
! [X32: paraco1604210848lle_fm,Y32: paraco1604210848lle_fm] :
( ( ( paraco445776468le_Neg @ X32 )
= ( paraco445776468le_Neg @ Y32 ) )
= ( X32 = Y32 ) ) ).
% fm.inject(2)
thf(fact_333_double__negation,axiom,
( paraco1325918602e_eval
= ( ^ [I: list_char > paraco1605129243lle_tv,P: paraco1604210848lle_fm] : ( paraco1325918602e_eval @ I @ ( paraco445776468le_Neg @ ( paraco445776468le_Neg @ P ) ) ) ) ) ).
% double_negation
thf(fact_334_fm_Odistinct_I21_J,axiom,
! [X32: paraco1604210848lle_fm,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X32 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_335_fm_Odistinct_I19_J,axiom,
! [X32: paraco1604210848lle_fm,X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X32 )
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_336_fm_Odistinct_I11_J,axiom,
! [X32: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco445776468le_Neg @ X32 ) ) ).
% fm.distinct(11)
thf(fact_337_eval__equality,axiom,
! [I2: list_char > paraco1605129243lle_tv,P3: paraco1604210848lle_fm,Q2: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1325918602e_eval @ I2 @ Q2 ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ Q2 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P3 )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ Q2 ) ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P3 )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P3 ) ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q2 )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P3 @ Q2 ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_equality
thf(fact_338_fm_Oexhaust,axiom,
! [Y3: paraco1604210848lle_fm] :
( ! [X12: list_char] :
( Y3
!= ( paraco1480028776le_Pro @ X12 ) )
=> ( ( Y3 != paraco367965286_Truth )
=> ( ! [X33: paraco1604210848lle_fm] :
( Y3
!= ( paraco445776468le_Neg @ X33 ) )
=> ( ! [X412: paraco1604210848lle_fm,X422: paraco1604210848lle_fm] :
( Y3
!= ( paraco69239110le_Con @ X412 @ X422 ) )
=> ( ! [X512: paraco1604210848lle_fm,X522: paraco1604210848lle_fm] :
( Y3
!= ( paraco1389086619le_Eql @ X512 @ X522 ) )
=> ~ ! [X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( Y3
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_339_fm_Oinduct,axiom,
! [P2: paraco1604210848lle_fm > $o,Fm: paraco1604210848lle_fm] :
( ! [X3: list_char] : ( P2 @ ( paraco1480028776le_Pro @ X3 ) )
=> ( ( P2 @ paraco367965286_Truth )
=> ( ! [X3: paraco1604210848lle_fm] :
( ( P2 @ X3 )
=> ( P2 @ ( paraco445776468le_Neg @ X3 ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco69239110le_Con @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco1389086619le_Eql @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco1745535428le_Eql @ X1a @ X22 ) ) ) )
=> ( P2 @ Fm ) ) ) ) ) ) ) ).
% fm.induct
thf(fact_340_fm_Oinject_I5_J,axiom,
! [X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm,Y61: paraco1604210848lle_fm,Y62: paraco1604210848lle_fm] :
( ( ( paraco1745535428le_Eql @ X612 @ X622 )
= ( paraco1745535428le_Eql @ Y61 @ Y62 ) )
= ( ( X612 = Y61 )
& ( X622 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_341_fm_Oinject_I1_J,axiom,
! [X1: list_char,Y1: list_char] :
( ( ( paraco1480028776le_Pro @ X1 )
= ( paraco1480028776le_Pro @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fm.inject(1)
thf(fact_342_fm_Odistinct_I9_J,axiom,
! [X1: list_char,X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(9)
thf(fact_343_eval_Osimps_I1_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,S2: list_char] :
( ( paraco1325918602e_eval @ I2 @ ( paraco1480028776le_Pro @ S2 ) )
= ( I2 @ S2 ) ) ).
% eval.simps(1)
thf(fact_344_fm_Odistinct_I23_J,axiom,
! [X32: paraco1604210848lle_fm,X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X32 )
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(23)
thf(fact_345_fm_Odistinct_I3_J,axiom,
! [X1: list_char,X32: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco445776468le_Neg @ X32 ) ) ).
% fm.distinct(3)
thf(fact_346_fm_Odistinct_I27_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( ( paraco69239110le_Con @ X41 @ X42 )
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(27)
thf(fact_347_fm_Odistinct_I29_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm,X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( ( paraco1389086619le_Eql @ X51 @ X52 )
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(29)
thf(fact_348_fm_Odistinct_I5_J,axiom,
! [X1: list_char,X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_349_fm_Odistinct_I7_J,axiom,
! [X1: list_char,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_350_fm_Odistinct_I17_J,axiom,
! [X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(17)
thf(fact_351_fm_Odistinct_I1_J,axiom,
! [X1: list_char] :
( ( paraco1480028776le_Pro @ X1 )
!= paraco367965286_Truth ) ).
% fm.distinct(1)
thf(fact_352_eval_Ocases,axiom,
! [X2: produc696645085lle_fm] :
( ! [I3: list_char > paraco1605129243lle_tv,S3: list_char] :
( X2
!= ( produc1710735575lle_fm @ I3 @ ( paraco1480028776le_Pro @ S3 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv] :
( X2
!= ( produc1710735575lle_fm @ I3 @ paraco367965286_Truth ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm] :
( X2
!= ( produc1710735575lle_fm @ I3 @ ( paraco445776468le_Neg @ P4 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X2
!= ( produc1710735575lle_fm @ I3 @ ( paraco69239110le_Con @ P4 @ Q3 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X2
!= ( produc1710735575lle_fm @ I3 @ ( paraco1389086619le_Eql @ P4 @ Q3 ) ) )
=> ~ ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X2
!= ( produc1710735575lle_fm @ I3 @ ( paraco1745535428le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_353_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
% Conjectures (1)
thf(conj_0,conjecture,
( ( ! [I: list_char > paraco1605129243lle_tv] :
( ( paraco1325918602e_eval @ I @ p )
= ( paraco1344940915le_Det @ $true ) ) )
!= ( ~ ! [I: list_char > paraco1605129243lle_tv] :
( ( ord_le241064091lle_tv @ ( image_1187894195lle_tv @ I @ top_to748628679t_char )
@ ( paraco1933902578domain
@ ( collect_nat
@ ^ [N: nat] : $true ) ) )
=> ( ( paraco1325918602e_eval @ I @ p )
= ( paraco1344940915le_Det @ $true ) ) ) ) ) ).
%------------------------------------------------------------------------------