TPTP Problem File: SEU299+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU299+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_xboole_0__e1_39_1__ordinal1
% 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__e1_39_1__ordinal1 [Urb07]
% Status : Theorem
% Rating : 0.64 v8.2.0, 0.61 v8.1.0, 0.53 v7.5.0, 0.69 v7.4.0, 0.53 v7.3.0, 0.55 v7.1.0, 0.52 v7.0.0, 0.57 v6.4.0, 0.58 v6.3.0, 0.62 v6.2.0, 0.64 v6.1.0, 0.70 v6.0.0, 0.74 v5.4.0, 0.79 v5.3.0, 0.81 v5.2.0, 0.70 v5.1.0, 0.76 v5.0.0, 0.83 v4.0.1, 0.87 v4.0.0, 0.88 v3.7.0, 0.85 v3.5.0, 0.84 v3.3.0
% Syntax : Number of formulae : 47 ( 11 unt; 0 def)
% Number of atoms : 186 ( 16 equ)
% Maximal formula atoms : 34 ( 3 avg)
% Number of connectives : 166 ( 27 ~; 0 |; 103 &)
% ( 2 <=>; 34 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-1 aty)
% Number of variables : 69 ( 44 !; 25 ?)
% 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__e18_27__finset_1__1,conjecture,
! [A] :
( ordinal(A)
=> ? [B] :
! [C] :
( in(C,B)
<=> ( in(C,succ(A))
& ? [D] :
( ordinal(D)
& C = D
& ( in(D,omega)
=> ! [E] :
( element(E,powerset(powerset(D)))
=> ~ ( E != empty_set
& ! [F] :
~ ( in(F,E)
& ! [G] :
( ( in(G,E)
& subset(F,G) )
=> G = F ) ) ) ) ) ) ) ) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(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_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(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
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(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(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(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(cc2_arytm_3,axiom,
! [A] :
( ( empty(A)
& ordinal(A) )
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(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(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(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(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_ordinal1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k5_ordinal2,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(fc1_ordinal2,axiom,
( epsilon_transitive(omega)
& epsilon_connected(omega)
& ordinal(omega)
& ~ empty(omega) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(cc1_arytm_3,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( element(B,A)
=> ( epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B) ) ) ) ).
fof(cc3_arytm_3,axiom,
! [A] :
( element(A,omega)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ) ).
fof(fc1_ordinal1,axiom,
! [A] : ~ empty(succ(A)) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
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(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(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(s1_tarski__e18_27__finset_1__1,axiom,
! [A] :
( ordinal(A)
=> ( ! [B,C,D] :
( ( B = C
& ? [E] :
( ordinal(E)
& C = E
& ( in(E,omega)
=> ! [F] :
( element(F,powerset(powerset(E)))
=> ~ ( F != empty_set
& ! [G] :
~ ( in(G,F)
& ! [H] :
( ( in(H,F)
& subset(G,H) )
=> H = G ) ) ) ) ) )
& B = D
& ? [I] :
( ordinal(I)
& D = I
& ( in(I,omega)
=> ! [J] :
( element(J,powerset(powerset(I)))
=> ~ ( J != empty_set
& ! [K] :
~ ( in(K,J)
& ! [L] :
( ( in(L,J)
& subset(K,L) )
=> L = K ) ) ) ) ) ) )
=> C = D )
=> ? [B] :
! [C] :
( in(C,B)
<=> ? [D] :
( in(D,succ(A))
& D = C
& ? [M] :
( ordinal(M)
& C = M
& ( in(M,omega)
=> ! [N] :
( element(N,powerset(powerset(M)))
=> ~ ( N != empty_set
& ! [O] :
~ ( in(O,N)
& ! [P] :
( ( in(P,N)
& subset(O,P) )
=> P = O ) ) ) ) ) ) ) ) ) ) ).
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