TPTP Problem File: SEU288+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU288+1 : TPTP v9.0.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s2_funct_1__e10_24__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-s2_funct_1__e10_24__wellord2 [Urb07]
% Status : Theorem
% Rating : 0.94 v8.2.0, 1.00 v7.5.0, 0.80 v7.4.0, 1.00 v7.3.0, 0.67 v7.0.0, 1.00 v6.4.0, 0.67 v6.2.0, 1.00 v5.4.0
% Syntax : Number of formulae : 46 ( 16 unt; 0 def)
% Number of atoms : 151 ( 19 equ)
% Maximal formula atoms : 25 ( 3 avg)
% Number of connectives : 122 ( 17 ~; 1 |; 67 &)
% ( 6 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 1 con; 0-2 aty)
% Number of variables : 82 ( 60 !; 22 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s2_funct_1__e10_24__wellord2,conjecture,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ( ( ! [C,D,E] :
( ( in(C,A)
& ? [F] :
( C = F
& in(D,F)
& ! [G] :
( in(G,F)
=> in(ordered_pair(D,G),B) ) )
& ? [H] :
( C = H
& in(E,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(E,I),B) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,A)
& ! [D] :
~ ? [J] :
( C = J
& in(D,J)
& ! [K] :
( in(K,J)
=> in(ordered_pair(D,K),B) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = A
& ! [D] :
( in(D,A)
=> ? [L] :
( D = L
& in(apply(C,D),L)
& ! [M] :
( in(M,L)
=> in(ordered_pair(apply(C,D),M),B) ) ) ) ) ) ) ).
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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(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(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(s1_funct_1__e10_24__wellord2__1,axiom,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ( ! [C,D,E] :
( ( in(C,A)
& ? [F] :
( C = F
& in(D,F)
& ! [G] :
( in(G,F)
=> in(ordered_pair(D,G),B) ) )
& in(C,A)
& ? [H] :
( C = H
& in(E,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(E,I),B) ) ) )
=> D = E )
=> ? [C] :
( relation(C)
& function(C)
& ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(D,A)
& in(D,A)
& ? [J] :
( D = J
& in(E,J)
& ! [K] :
( in(K,J)
=> in(ordered_pair(E,K),B) ) ) ) ) ) ) ) ).
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(d4_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( ( in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> in(ordered_pair(B,C),A) ) )
& ( ~ in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> C = empty_set ) ) ) ) ).
fof(d4_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( B = relation_dom(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] : in(ordered_pair(C,D),A) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k2_tarski,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(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
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(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(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) ).
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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