## TPTP Problem File: SEU298+1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU298+1 : TPTP v8.1.2. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem s1_xboole_0__e4_27_3_1__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__e4_27_3_1__finset_1 [Urb07]

% Status   : Theorem
% Rating   : 0.42 v8.1.0, 0.39 v7.5.0, 0.44 v7.4.0, 0.30 v7.3.0, 0.34 v7.2.0, 0.31 v7.1.0, 0.43 v7.0.0, 0.37 v6.4.0, 0.38 v6.3.0, 0.46 v6.2.0, 0.48 v6.1.0, 0.57 v6.0.0, 0.52 v5.5.0, 0.56 v5.4.0, 0.57 v5.3.0, 0.59 v5.2.0, 0.40 v5.1.0, 0.43 v5.0.0, 0.54 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    :   42 (  10 unt;   0 def)
%            Number of atoms       :  139 (   8 equ)
%            Maximal formula atoms :   14 (   3 avg)
%            Number of connectives :  113 (  16   ~;   0   |;  72   &)
%                                         (   2 <=>;  23  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   5 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :   13 (  11 usr;   1 prp; 0-2 aty)
%            Number of functors    :    4 (   4 usr;   0 con; 1-2 aty)
%            Number of variables   :   60 (  37   !;  23   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_xboole_0__e4_27_3_1__finset_1,conjecture,
! [A,B] :
( ( ordinal(A)
& element(B,powerset(powerset(succ(A)))) )
=> ? [C] :
! [D] :
( in(D,C)
<=> ( in(D,powerset(A))
& ? [E] :
( in(E,B)
& D = set_difference(E,singleton(A)) ) ) ) ) ).

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(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(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(rc1_funct_1,axiom,
? [A] :
( relation(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(fc3_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_difference(A,B)) ) ).

fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).

fof(rc1_arytm_3,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ).

fof(fc2_arytm_3,axiom,
! [A] :
( ( ordinal(A)
& natural(A) )
=> ( ~ empty(succ(A))
& epsilon_transitive(succ(A))
& epsilon_connected(succ(A))
& ordinal(succ(A))
& natural(succ(A)) ) ) ).

fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).

fof(cc1_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(fc12_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_difference(A,B)) ) ).

fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).

fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).

fof(cc2_arytm_3,axiom,
! [A] :
( ( empty(A)
& ordinal(A) )
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(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(cc3_ordinal1,axiom,
! [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(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_ordinal1,axiom,
\$true ).

fof(dt_k1_tarski,axiom,
\$true ).

fof(dt_k1_zfmisc_1,axiom,
\$true ).

fof(dt_k4_xboole_0,axiom,
\$true ).

fof(dt_m1_subset_1,axiom,
\$true ).

fof(fc1_finset_1,axiom,
! [A] :
( ~ empty(singleton(A))
& finite(singleton(A)) ) ).

fof(cc1_arytm_3,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( element(B,A)
=> ( epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B) ) ) ) ).

fof(fc1_ordinal1,axiom,
! [A] : ~ empty(succ(A)) ).

fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).

fof(fc3_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( ~ empty(succ(A))
& epsilon_transitive(succ(A))
& epsilon_connected(succ(A))
& ordinal(succ(A)) ) ) ).

fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).

fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).

fof(s1_tarski__e4_27_3_1__finset_1__1,axiom,
! [A,B] :
( ( ordinal(A)
& element(B,powerset(powerset(succ(A)))) )
=> ( ! [C,D,E] :
( ( C = D
& ? [F] :
( in(F,B)
& D = set_difference(F,singleton(A)) )
& C = E
& ? [G] :
( in(G,B)
& E = set_difference(G,singleton(A)) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,powerset(A))
& E = D
& ? [H] :
( in(H,B)
& D = set_difference(H,singleton(A)) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```