TPTP Problem File: SEU285+1.p
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%------------------------------------------------------------------------------
% File : SEU285+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t26_wellord2
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-t26_wellord2 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.3.0
% Syntax : Number of formulae : 80 ( 29 unt; 0 def)
% Number of atoms : 208 ( 27 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 145 ( 17 ~; 2 |; 69 &)
% ( 14 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-2 aty)
% Number of functors : 14 ( 14 usr; 1 con; 0-2 aty)
% Number of variables : 127 ( 111 !; 16 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(cc2_ordinal1,axiom,
! [A] :
( ( epsilon_transitive(A)
& epsilon_connected(A) )
=> ordinal(A) ) ).
fof(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(connectedness_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
| ordinal_subset(B,A) ) ) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d1_wellord2,axiom,
! [A,B] :
( relation(B)
=> ( B = inclusion_relation(A)
<=> ( relation_field(B) = A
& ! [C,D] :
( ( in(C,A)
& in(D,A) )
=> ( in(ordered_pair(C,D),B)
<=> subset(C,D) ) ) ) ) ) ).
fof(d2_ordinal1,axiom,
! [A] :
( epsilon_transitive(A)
<=> ! [B] :
( in(B,A)
=> subset(B,A) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
fof(d4_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
<=> ? [C] :
( relation(C)
& function(C)
& one_to_one(C)
& relation_dom(C) = A
& relation_rng(C) = B ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d6_relat_1,axiom,
! [A] :
( relation(A)
=> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)) ) ).
fof(d6_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] : relation_restriction(A,B) = set_intersection2(A,cartesian_product2(B,B)) ) ).
fof(d8_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
<=> ! [B,C] :
( ( in(B,relation_dom(A))
& in(C,relation_dom(A))
& apply(A,B) = apply(A,C) )
=> B = C ) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_wellord2,axiom,
! [A] : relation(inclusion_relation(A)) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_wellord1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_restriction(A,B)) ) ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_relat_1,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc2_ordinal1,axiom,
( relation(empty_set)
& relation_empty_yielding(empty_set)
& function(empty_set)
& one_to_one(empty_set)
& empty(empty_set)
& epsilon_transitive(empty_set)
& epsilon_connected(empty_set)
& ordinal(empty_set) ) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(l30_wellord2,axiom,
! [A,B] :
( relation(B)
=> ~ ( well_ordering(B)
& equipotent(A,relation_field(B))
& ! [C] :
( relation(C)
=> ~ well_orders(C,A) ) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_ordinal1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A)
& empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_ordinal1,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(A) ) ).
fof(redefinition_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
<=> subset(A,B) ) ) ).
fof(redefinition_r2_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
<=> are_equipotent(A,B) ) ).
fof(reflexivity_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ordinal_subset(A,A) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(reflexivity_r2_wellord2,axiom,
! [A,B] : equipotent(A,A) ).
fof(s1_xboole_0__e6_22__wellord2,axiom,
! [A] :
? [B] :
! [C] :
( in(C,B)
<=> ( in(C,A)
& ordinal(C) ) ) ).
fof(s3_funct_1__e16_22__wellord2,axiom,
! [A] :
? [B] :
( relation(B)
& function(B)
& relation_dom(B) = A
& ! [C] :
( in(C,A)
=> apply(B,C) = singleton(C) ) ) ).
fof(symmetry_r2_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
=> equipotent(B,A) ) ).
fof(t136_zfmisc_1,axiom,
! [A] :
? [B] :
( in(A,B)
& ! [C,D] :
( ( in(C,B)
& subset(D,C) )
=> in(D,B) )
& ! [C] :
( in(C,B)
=> in(powerset(C),B) )
& ! [C] :
~ ( subset(C,B)
& ~ are_equipotent(C,B)
& ~ in(C,B) ) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t23_ordinal1,axiom,
! [A,B] :
( ordinal(B)
=> ( in(A,B)
=> ordinal(A) ) ) ).
fof(t25_wellord2,axiom,
! [A,B] :
( relation(B)
=> ( well_orders(B,A)
=> ( relation_field(relation_restriction(B,A)) = A
& well_ordering(relation_restriction(B,A)) ) ) ) ).
fof(t26_wellord2,conjecture,
! [A] :
? [B] :
( relation(B)
& well_orders(B,A) ) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t31_ordinal1,axiom,
! [A] :
( ! [B] :
( in(B,A)
=> ( ordinal(B)
& subset(B,A) ) )
=> ordinal(A) ) ).
fof(t32_wellord1,axiom,
! [A,B] :
( relation(B)
=> ( well_ordering(B)
=> well_ordering(relation_restriction(B,A)) ) ) ).
fof(t39_wellord1,axiom,
! [A,B] :
( relation(B)
=> ( ( well_ordering(B)
& subset(A,relation_field(B)) )
=> relation_field(relation_restriction(B,A)) = A ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t6_zfmisc_1,axiom,
! [A,B] :
( subset(singleton(A),singleton(B))
=> A = B ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t7_wellord2,axiom,
! [A] :
( ordinal(A)
=> well_ordering(inclusion_relation(A)) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
%------------------------------------------------------------------------------