TPTP Problem File: ITP029^2.p
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%------------------------------------------------------------------------------
% File : ITP029^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_163__3251696_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 : BinaryTree/prob_163__3251696_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 347 ( 136 unt; 54 typ; 0 def)
% Number of atoms : 802 ( 284 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3296 ( 86 ~; 31 |; 60 &;2752 @)
% ( 0 <=>; 367 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 204 ( 204 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 7 con; 0-6 aty)
% Number of variables : 1031 ( 91 ^; 847 !; 46 ?;1031 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:27.169
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
binary1291135688e_Tree: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (50)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
binary210054475elle_T:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
binary1746293266le_Tip:
!>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Ocase__Tree,type,
binary536355927e_Tree:
!>[B: $tType,A: $tType] : ( B > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B ) > ( binary1291135688e_Tree @ A ) > B ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Orec__Tree,type,
binary1929596613c_Tree:
!>[C: $tType,A: $tType] : ( C > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C ) > ( binary1291135688e_Tree @ A ) > C ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Oset__Tree,type,
binary2130109271t_Tree:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oeqs,type,
binary64540844le_eqs:
!>[A: $tType] : ( ( A > int ) > A > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
binary1653327646_setOf:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
binary1610619414edTree:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
binary231205461t_pred:
!>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Zorn_Ochains,type,
chains:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t1____,type,
t1: binary1291135688e_Tree @ a ).
thf(sy_v_t2____,type,
t2: binary1291135688e_Tree @ a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (256)
thf(fact_0_hahb,axiom,
( ( h @ a2 )
= ( h @ b ) ) ).
% hahb
thf(fact_1__092_060open_062h_Ab_A_060_Ah_Aa_092_060close_062,axiom,
ord_less @ int @ ( h @ b ) @ ( h @ a2 ) ).
% \<open>h b < h a\<close>
thf(fact_2__092_060open_062a_A_061_Ax_A_092_060Longrightarrow_062_Aa_A_061_Ab_092_060close_062,axiom,
( ( a2 = x )
=> ( a2 = b ) ) ).
% \<open>a = x \<Longrightarrow> a = b\<close>
thf(fact_3_adef1,axiom,
member @ a @ a2 @ ( binary1653327646_setOf @ a @ t2 ) ).
% adef1
thf(fact_4_bdef1,axiom,
member @ a @ b @ ( binary1653327646_setOf @ a @ t1 ) ).
% bdef1
thf(fact_5__092_060open_062a_A_092_060in_062_AsetOf_At1_A_092_060Longrightarrow_062_Aa_A_061_Ab_092_060close_062,axiom,
( ( member @ a @ a2 @ ( binary1653327646_setOf @ a @ t1 ) )
=> ( a2 = b ) ) ).
% \<open>a \<in> setOf t1 \<Longrightarrow> a = b\<close>
thf(fact_6_o1,axiom,
ord_less @ int @ ( h @ b ) @ ( h @ x ) ).
% o1
thf(fact_7_o2,axiom,
ord_less @ int @ ( h @ x ) @ ( h @ a2 ) ).
% o2
thf(fact_8_h2,axiom,
binary231205461t_pred @ a @ h @ a2 @ b @ t2 ).
% h2
thf(fact_9_h1,axiom,
binary231205461t_pred @ a @ h @ a2 @ b @ t1 ).
% h1
thf(fact_10__092_060open_062sorted__distinct__pred_Ah_Aa_Ab_ATip_092_060close_062,axiom,
binary231205461t_pred @ a @ h @ a2 @ b @ ( binary1746293266le_Tip @ a ) ).
% \<open>sorted_distinct_pred h a b Tip\<close>
thf(fact_11_calculation,axiom,
( ( member @ a @ b @ ( binary1653327646_setOf @ a @ t1 ) )
| ( b = x )
| ( member @ a @ b @ ( binary1653327646_setOf @ a @ t2 ) ) ) ).
% calculation
thf(fact_12__092_060open_062a_A_092_060in_062_AsetOf_At1_A_092_060or_062_Aa_A_061_Ax_A_092_060or_062_Aa_A_092_060in_062_AsetOf_At2_092_060close_062,axiom,
( ( member @ a @ a2 @ ( binary1653327646_setOf @ a @ t1 ) )
| ( a2 = x )
| ( member @ a @ a2 @ ( binary1653327646_setOf @ a @ t2 ) ) ) ).
% \<open>a \<in> setOf t1 \<or> a = x \<or> a \<in> setOf t2\<close>
thf(fact_13_bdef,axiom,
member @ a @ b @ ( binary1653327646_setOf @ a @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ).
% bdef
thf(fact_14_Tree_Oinject,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
= ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_15_adef,axiom,
member @ a @ a2 @ ( binary1653327646_setOf @ a @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ) ).
% adef
thf(fact_16_s,axiom,
binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% s
thf(fact_17_Tree_Odistinct_I1_J,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary1746293266le_Tip @ A )
!= ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_18_Tree_Oinduct,axiom,
! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
( ( P @ ( binary1746293266le_Tip @ A ) )
=> ( ! [X1: binary1291135688e_Tree @ A,X2: A,X3: binary1291135688e_Tree @ A] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( binary210054475elle_T @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_19_Tree_Oexhaust,axiom,
! [A: $tType,Y: binary1291135688e_Tree @ A] :
( ( Y
!= ( binary1746293266le_Tip @ A ) )
=> ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
( Y
!= ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_20_s2,axiom,
binary1610619414edTree @ a @ h @ t2 ).
% s2
thf(fact_21_s1,axiom,
binary1610619414edTree @ a @ h @ t1 ).
% s1
thf(fact_22_sortedTree_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( ( binary1610619414edTree @ A @ H @ T1 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
=> ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
=> ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
& ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_23_sorted__distinct__pred__def,axiom,
! [A: $tType] :
( ( binary231205461t_pred @ A )
= ( ^ [H2: A > int,A2: A,B2: A,T: binary1291135688e_Tree @ A] :
( ( ( binary1610619414edTree @ A @ H2 @ T )
& ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T ) )
& ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T ) )
& ( ( H2 @ A2 )
= ( H2 @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_24_eqs__def,axiom,
! [A: $tType] :
( ( binary64540844le_eqs @ A )
= ( ^ [H2: A > int,X4: A] :
( collect @ A
@ ^ [Y2: A] :
( ( H2 @ Y2 )
= ( H2 @ X4 ) ) ) ) ) ).
% eqs_def
thf(fact_25_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X5: C] :
( ( ord_less @ C @ X5 @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_26_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less @ A @ T3 @ X5 ) ) ) ).
% minf(7)
thf(fact_27_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less @ A @ X5 @ T3 ) ) ) ).
% minf(5)
thf(fact_28_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T3 ) ) ) ).
% minf(4)
thf(fact_29_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T3 ) ) ) ).
% minf(3)
thf(fact_30_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_31_sortLemmaR,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_32_sortLemmaL,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_33_sortedTree_Osimps_I1_J,axiom,
! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% sortedTree.simps(1)
thf(fact_34_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_35_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_36_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T3 ) ) ) ).
% pinf(3)
thf(fact_37_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T3 ) ) ) ).
% pinf(4)
thf(fact_38_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less @ A @ X5 @ T3 ) ) ) ).
% pinf(5)
thf(fact_39_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less @ A @ T3 @ X5 ) ) ) ).
% pinf(7)
thf(fact_40_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X5: C] :
( ( ord_less @ C @ Z @ X5 )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_41_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( P @ X6 )
= ( P2 @ X6 ) ) )
=> ( ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_42_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( ord_less @ A @ B3 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_43_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_44_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A] :
( ( ord_less @ A @ B4 @ A3 )
=> ( A3 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X6: A] :
( ( F2 @ X6 )
= ( G @ X6 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_49_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_50_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X5: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% linordered_field_no_lb
thf(fact_51_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less @ A @ A3 @ B4 )
=> ( A3 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_52_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R: B,F3: A > B,G2: A > B,X6: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X6 )
=> ( ( F3 @ Y4 )
= ( G2 @ Y4 ) ) )
=> ( ( P @ F3 @ X6 @ R )
= ( P @ G2 @ X6 @ R ) ) )
=> ( ! [X6: A,F3: A > B] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X6 )
=> ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
=> ? [X_12: B] : ( P @ F3 @ X6 @ X_12 ) )
=> ? [F3: A > B] :
! [X5: A] : ( P @ F3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_53_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_54_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A,C2: A] :
( ( ord_less @ A @ B4 @ A3 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_55_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C2: B] :
( ( A3
= ( F2 @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X6: B,Y3: B] :
( ( ord_less @ B @ X6 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X6 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_56_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B4: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( ( F2 @ B4 )
= C2 )
=> ( ! [X6: A,Y3: A] :
( ( ord_less @ A @ X6 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X6 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_57_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X6: B,Y3: B] :
( ( ord_less @ B @ X6 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X6 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_58_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B4: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( ord_less @ C @ ( F2 @ B4 ) @ C2 )
=> ( ! [X6: A,Y3: A] :
( ( ord_less @ A @ X6 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X6 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_59_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_60_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_61_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_62_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_63_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less @ A @ A3 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A3 ) ) ) ).
% order.asym
thf(fact_64_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_65_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_66_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_67_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B4: A] :
( ( ord_less @ A @ A3 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A3 ) ) ) ).
% less_asym'
thf(fact_68_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% less_trans
thf(fact_69_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_70_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_71_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( A3 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_72_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_73_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A] :
( ( ord_less @ A @ B4 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_74_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_75_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_76_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X6: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X6 )
=> ( P @ Y4 ) )
=> ( P @ X6 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_77_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_78_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_79_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_80_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_81_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_82_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_83_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_84_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X7: A] : ( P3 @ X7 ) )
= ( ^ [P4: A > $o] :
? [N: A] :
( ( P4 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_85_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B3: A] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A3 @ B4 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_86_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_87_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X6: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X6 ) )
=> ( P @ Y4 ) )
=> ( P @ X6 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_88_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X6: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X6 ) )
=> ( P @ Y4 ) )
=> ( P @ X6 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_89_Tree_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B] :
( ( binary536355927e_Tree @ B @ A @ F1 @ F22 @ ( binary1746293266le_Tip @ A ) )
= F1 ) ).
% Tree.simps(4)
thf(fact_90_Tree_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary536355927e_Tree @ B @ A @ F1 @ F22 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 ) ) ).
% Tree.simps(5)
thf(fact_91_setOf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% setOf.simps(1)
thf(fact_92_Tree_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C] :
( ( binary1929596613c_Tree @ C @ A @ F1 @ F22 @ ( binary1746293266le_Tip @ A ) )
= F1 ) ).
% Tree.simps(6)
thf(fact_93_Tree_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary1929596613c_Tree @ C @ A @ F1 @ F22 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 @ ( binary1929596613c_Tree @ C @ A @ F1 @ F22 @ X21 ) @ ( binary1929596613c_Tree @ C @ A @ F1 @ F22 @ X23 ) ) ) ).
% Tree.simps(7)
thf(fact_94_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_95_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_96_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_97_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_98_Tree_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,Tree: binary1291135688e_Tree @ A] :
( ( H @ ( binary536355927e_Tree @ B @ A @ F1 @ F22 @ Tree ) )
= ( binary536355927e_Tree @ C @ A @ ( H @ F1 )
@ ^ [X12: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] : ( H @ ( F22 @ X12 @ X24 @ X32 ) )
@ Tree ) ) ).
% Tree.case_distrib
thf(fact_99_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_100_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_101_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_102_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_103_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_104_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_105_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_106_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_107_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_108_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_109_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_110_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_111_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_112_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_113_Tree_Osimps_I14_J,axiom,
! [A: $tType] :
( ( binary2130109271t_Tree @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Tree.simps(14)
thf(fact_114_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,Y3: A] :
( ( member @ A @ X6 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( X6 = Y3 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_115_psubsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_116_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_117_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_118_Tree_Oset__cases,axiom,
! [A: $tType,E: A,A3: binary1291135688e_Tree @ A] :
( ( member @ A @ E @ ( binary2130109271t_Tree @ A @ A3 ) )
=> ( ! [Z1: binary1291135688e_Tree @ A] :
( ? [Z22: A,Z32: binary1291135688e_Tree @ A] :
( A3
= ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z1 ) ) )
=> ( ! [Z1: binary1291135688e_Tree @ A,Z32: binary1291135688e_Tree @ A] :
( A3
!= ( binary210054475elle_T @ A @ Z1 @ E @ Z32 ) )
=> ~ ! [Z1: binary1291135688e_Tree @ A,Z22: A,Z32: binary1291135688e_Tree @ A] :
( ( A3
= ( binary210054475elle_T @ A @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member @ A @ E @ ( binary2130109271t_Tree @ A @ Z32 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_119_Tree_Oset__intros_I1_J,axiom,
! [A: $tType,Y: A,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ X21 ) )
=> ( member @ A @ Y @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_120_Tree_Oset__intros_I2_J,axiom,
! [A: $tType,X22: A,X21: binary1291135688e_Tree @ A,X23: binary1291135688e_Tree @ A] : ( member @ A @ X22 @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(2)
thf(fact_121_Tree_Oset__intros_I3_J,axiom,
! [A: $tType,Ya: A,X23: binary1291135688e_Tree @ A,X21: binary1291135688e_Tree @ A,X22: A] :
( ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ X23 ) )
=> ( member @ A @ Ya @ ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_122_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_123_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X4: A] :
( A6
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_124_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X6: A] :
( A4
!= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_125_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R2: B > B > $o,F4: A > B,X4: A,Y2: A] : ( R2 @ ( F4 @ X4 ) @ ( F4 @ Y2 ) ) ) ) ).
% inv_imagep_def
thf(fact_126_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_127_insert__iff,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
= ( ( A3 = B4 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_128_insertCI,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( ~ ( member @ A @ A3 @ B5 )
=> ( A3 = B4 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertCI
thf(fact_129_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_130_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_131_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_132_singleton__inject,axiom,
! [A: $tType,A3: A,B4: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_133_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_134_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B4: A,C2: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B4 = D2 ) )
| ( ( A3 = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_135_singleton__iff,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_136_singletonD,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_137_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_138_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_139_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A3 @ B7 ) )
& ~ ( member @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_140_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_141_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A,B5: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B4 @ B5 )
=> ( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ B5 ) )
= ( ( ( A3 = B4 )
=> ( A4 = B5 ) )
& ( ( A3 != B4 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B4 @ C4 ) )
& ~ ( member @ A @ B4 @ C4 )
& ( B5
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_142_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_143_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_144_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_145_insertI2,axiom,
! [A: $tType,A3: A,B5: set @ A,B4: A] :
( ( member @ A @ A3 @ B5 )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% insertI2
thf(fact_146_insertI1,axiom,
! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% insertI1
thf(fact_147_insertE,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
=> ( ( A3 != B4 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_148_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_149_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_150_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_151_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_152_setOf_Osimps_I2_J,axiom,
! [A: $tType,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1653327646_setOf @ A @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary1653327646_setOf @ A @ T1 ) @ ( binary1653327646_setOf @ A @ T2 ) ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% setOf.simps(2)
thf(fact_153_Tree_Osimps_I15_J,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary2130109271t_Tree @ A @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( binary2130109271t_Tree @ A @ X21 ) @ ( insert @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( binary2130109271t_Tree @ A @ X23 ) ) ) ).
% Tree.simps(15)
thf(fact_154_UnCI,axiom,
! [A: $tType,C2: A,B5: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnCI
thf(fact_155_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_156_Un__empty,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_157_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A3: A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B5 ) )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_158_Un__insert__left,axiom,
! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_159_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_160_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI1
thf(fact_161_UnI2,axiom,
! [A: $tType,C2: A,B5: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% UnI2
thf(fact_162_bex__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_163_ball__Un,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_164_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_165_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_166_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_167_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B5 ) ) ).
% Un_left_absorb
thf(fact_168_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_169_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_170_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
| ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_171_Un__empty__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_172_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_173_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A2 ) ) ) ) ).
% insert_def
thf(fact_174_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A2: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_175_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B5 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_176_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B5 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_177_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_178_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_179_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_180_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_181_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A,B4: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B4 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B4
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_182_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_183_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A3: A,B4: A] :
( ( ( sup_sup @ A @ A3 @ B4 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B4
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_184_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_185_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_186_sup__Un__eq,axiom,
! [A: $tType,R3: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R3 )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_187_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B4: A,A3: A] :
( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_188_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B4: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_189_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_190_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B4 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B4 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_191_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B4: A,A3: A] :
( ( ord_less @ A @ X @ B4 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% less_supI2
thf(fact_192_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B4: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B4 ) ) ) ) ).
% less_supI1
thf(fact_193_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X4: A] :
( X8
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_194_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C3: set @ A,Z3: A] :
( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( ( member @ A @ Z3 @ A4 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ C3 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X6
@ Z3 ) )
=> ( pred_chain @ A @ A4 @ P @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_195_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_196_subset_Ochain__extend,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),Z3: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ Z3 @ A4 )
=> ( ! [X6: set @ A] :
( ( member @ ( set @ A ) @ X6 @ C3 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X6
@ Z3 ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_197_subset_Ochain__total,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ X @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_198_pred__on_Ochain__total,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,C3: set @ A,X: A,Y: A] :
( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( ( member @ A @ X @ C3 )
=> ( ( member @ A @ Y @ C3 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_199_subset_Ochain__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_200_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o] : ( pred_chain @ A @ A4 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_201_chain__mono,axiom,
! [A: $tType,A4: set @ A,P: A > A > $o,Q: A > A > $o,C3: set @ A] :
( ! [X6: A,Y3: A] :
( ( member @ A @ X6 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( P @ X6 @ Y3 )
=> ( Q @ X6 @ Y3 ) ) ) )
=> ( ( pred_chain @ A @ A4 @ P @ C3 )
=> ( pred_chain @ A @ A4 @ Q @ C3 ) ) ) ).
% chain_mono
thf(fact_202_pred__on_Ochain_Ocong,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( pred_chain @ A ) ) ).
% pred_on.chain.cong
thf(fact_203_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X6: A] :
( ( P @ X6 )
=> ( X6 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_204_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_205_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_206_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_207_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ! [X6: A] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_208_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X4: A,Y2: A] :
( the @ A
@ ^ [Z5: A] :
( ( P4
=> ( Z5 = X4 ) )
& ( ~ P4
=> ( Z5 = Y2 ) ) ) ) ) ) ).
% If_def
thf(fact_209_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X6: A] :
( ( P @ X6 )
=> ( X6 = A3 ) )
=> ( ! [X6: A] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_210_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_211_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X6: A] :
( ( P @ X6 )
=> ( X6 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_212_chains__alt__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [A6: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_213_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_214_Pow__singleton__iff,axiom,
! [A: $tType,X9: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X9 )
= ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X9
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_215_Pow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( pow @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_216_Pow__bottom,axiom,
! [A: $tType,B5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B5 ) ) ).
% Pow_bottom
thf(fact_217_Pow__top,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow @ A @ A4 ) ) ).
% Pow_top
thf(fact_218_chains__extend,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A ),Z3: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S ) )
=> ( ( member @ ( set @ A ) @ Z3 @ S )
=> ( ! [X6: set @ A] :
( ( member @ ( set @ A ) @ X6 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ Z3 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_219_Powp__Pow__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( powp @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( ^ [X4: set @ A] : ( member @ ( set @ A ) @ X4 @ ( pow @ A @ A4 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_220_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_221_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_222_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( member @ A @ X6 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_223_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_224_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_225_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_226_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B5 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_227_psubsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_228_PowI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B5 ) ) ) ).
% PowI
thf(fact_229_Pow__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% Pow_iff
thf(fact_230_singleton__insert__inj__eq,axiom,
! [A: $tType,B4: A,A3: A,A4: set @ A] :
( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A4 ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_231_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A] :
( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_232_PowD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% PowD
thf(fact_233_Pow__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A4 ) @ ( pow @ A @ B5 ) ) ) ).
% Pow_mono
thf(fact_234_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A6: set @ A] :
( collect @ ( set @ A )
@ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% Pow_def
thf(fact_235_chainsD,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S ) )
=> ( ( member @ ( set @ A ) @ X @ C2 )
=> ( ( member @ ( set @ A ) @ Y @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_236_chainsD2,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S ) )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ C2 @ S ) ) ).
% chainsD2
thf(fact_237_Zorn__Lemma2,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [X6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X6 @ ( chains @ A @ A4 ) )
=> ? [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
& ! [Xb: set @ A] :
( ( member @ ( set @ A ) @ Xb @ X6 )
=> ( ord_less_eq @ ( set @ A ) @ Xb @ Xa ) ) ) )
=> ? [X6: set @ A] :
( ( member @ ( set @ A ) @ X6 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ Xa )
=> ( Xa = X6 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_238_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B4: A] :
( ( A3 = B4 )
| ~ ( ord_less_eq @ A @ A3 @ B4 )
| ~ ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_239_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_240_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_241_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_242_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A,B3: A] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_243_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_244_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_245_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_246_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_247_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_250_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_251_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_252_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_253_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_254_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_255_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
% Type constructors (33)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_4,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_5,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_6,axiom,
ord @ int ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_7,axiom,
! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_8,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_9,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_11,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_12,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_13,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_14,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_15,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_16,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_18,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_20,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_21,axiom,
bot @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
a2 = b ).
%------------------------------------------------------------------------------