TPTP Problem File: SEU282+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU282+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_funct_1__e2_11_1__funct_1
% 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-s1_funct_1__e2_11_1__funct_1 [Urb07]
% Status : Theorem
% Rating : 0.92 v8.1.0, 0.89 v7.5.0, 0.97 v7.4.0, 0.90 v7.3.0, 0.93 v7.1.0, 0.87 v7.0.0, 0.97 v6.4.0, 0.96 v6.1.0, 1.00 v6.0.0, 0.96 v5.5.0, 1.00 v5.1.0, 0.95 v5.0.0, 0.96 v3.7.0, 0.90 v3.5.0, 0.95 v3.3.0
% Syntax : Number of formulae : 46 ( 15 unt; 0 def)
% Number of atoms : 123 ( 19 equ)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 92 ( 15 ~; 1 |; 53 &)
% ( 6 <=>; 17 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 1 con; 0-2 aty)
% Number of variables : 77 ( 59 !; 18 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_funct_1__e16_22__wellord2__1,conjecture,
! [A] :
( ! [B,C,D] :
( ( in(B,A)
& C = singleton(B)
& in(B,A)
& D = singleton(B) )
=> C = D )
=> ? [B] :
( relation(B)
& function(B)
& ! [C,D] :
( in(ordered_pair(C,D),B)
<=> ( in(C,A)
& in(C,A)
& D = singleton(C) ) ) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
fof(cc2_ordinal1,axiom,
! [A] :
( ( epsilon_transitive(A)
& epsilon_connected(A) )
=> ordinal(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(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(rc3_ordinal1,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(s1_tarski__e16_22__wellord2__1,axiom,
! [A] :
( ! [B,C,D] :
( ( in(B,A)
& C = singleton(B)
& in(B,A)
& D = singleton(B) )
=> C = D )
=> ? [B] :
! [C] :
( in(C,B)
<=> ? [D] :
( in(D,A)
& in(D,A)
& C = singleton(D) ) ) ) ).
fof(s1_xboole_0__e16_22__wellord2__1,axiom,
! [A,B] :
? [C] :
! [D] :
( in(D,C)
<=> ( in(D,cartesian_product2(A,B))
& ? [E,F] :
( ordered_pair(E,F) = D
& in(E,A)
& F = singleton(E) ) ) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(d1_funct_1,axiom,
! [A] :
( function(A)
<=> ! [B,C,D] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(B,D),A) )
=> C = D ) ) ).
fof(d1_relat_1,axiom,
! [A] :
( relation(A)
<=> ! [B] :
~ ( in(B,A)
& ! [C,D] : B != ordered_pair(C,D) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(t106_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(A,C)
& in(B,D) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t33_zfmisc_1,axiom,
! [A,B,C,D] :
( ordered_pair(A,B) = ordered_pair(C,D)
=> ( A = C
& B = D ) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
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