TPTP Problem File: SEU297+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU297+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_xboole_0__e6_27__finset_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_xboole_0__e6_27__finset_1 [Urb07]
% Status : Theorem
% Rating : 0.47 v8.2.0, 0.44 v8.1.0, 0.42 v7.5.0, 0.47 v7.4.0, 0.33 v7.3.0, 0.38 v7.2.0, 0.34 v7.1.0, 0.43 v7.0.0, 0.37 v6.4.0, 0.38 v6.3.0, 0.50 v6.2.0, 0.48 v6.1.0, 0.57 v5.5.0, 0.56 v5.4.0, 0.61 v5.3.0, 0.59 v5.2.0, 0.40 v5.1.0, 0.43 v5.0.0, 0.50 v4.1.0, 0.52 v4.0.1, 0.57 v4.0.0, 0.58 v3.7.0, 0.60 v3.5.0, 0.68 v3.4.0, 0.63 v3.3.0
% Syntax : Number of formulae : 37 ( 7 unt; 0 def)
% Number of atoms : 125 ( 4 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 101 ( 13 ~; 0 |; 64 &)
% ( 2 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-2 aty)
% Number of functors : 3 ( 3 usr; 0 con; 1-2 aty)
% Number of variables : 53 ( 34 !; 19 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
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fof(s1_xboole_0__e6_27__finset_1,conjecture,
! [A,B,C] :
( ( element(B,powerset(powerset(A)))
& relation(C)
& function(C) )
=> ? [D] :
! [E] :
( in(E,D)
<=> ( in(E,powerset(relation_dom(C)))
& in(relation_image(C,E),B) ) ) ) ).
fof(rc2_finset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B)
& relation(B)
& function(B)
& one_to_one(B)
& epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B)
& natural(B)
& finite(B) ) ).
fof(rc1_arytm_3,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(fc13_finset_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& finite(B) )
=> finite(relation_image(A,B)) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(cc1_arytm_3,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( element(B,A)
=> ( epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B) ) ) ) ).
fof(cc2_arytm_3,axiom,
! [A] :
( ( empty(A)
& ordinal(A) )
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(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_finset_1,axiom,
! [A] :
( empty(A)
=> finite(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(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
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(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
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_relat_1,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(s1_tarski__e6_27__finset_1__1,axiom,
! [A,B,C] :
( ( element(B,powerset(powerset(A)))
& relation(C)
& function(C) )
=> ( ! [D,E,F] :
( ( D = E
& in(relation_image(C,E),B)
& D = F
& in(relation_image(C,F),B) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,powerset(relation_dom(C)))
& F = E
& in(relation_image(C,E),B) ) ) ) ) ).
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