TPTP Problem File: NUM382+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM382+1 : TPTP v9.0.0. Released v3.2.0.
% Domain : Number Theory (Ordinals)
% Problem : Ordinal numbers, theorem 6
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [Ban90] Bancerek (1990), The Ordinal Numbers
% [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : ordinal1__t6_ordinal1 [Urb06]
% Status : Theorem
% Rating : 0.94 v8.2.0, 0.97 v8.1.0, 0.89 v7.5.0, 0.97 v7.1.0, 0.96 v7.0.0, 1.00 v4.0.1, 0.96 v3.7.0, 0.85 v3.5.0, 0.89 v3.4.0, 1.00 v3.2.0
% Syntax : Number of formulae : 26 ( 4 unt; 0 def)
% Number of atoms : 71 ( 9 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 62 ( 17 ~; 1 |; 35 &)
% ( 2 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-2 aty)
% Number of functors : 2 ( 2 usr; 1 con; 0-6 aty)
% Number of variables : 44 ( 33 !; 11 ?)
% 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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(d4_enumset1,axiom,
! [A,B,C,D,E,F,G] :
( G = unordered_sextuple(A,B,C,D,E,F)
<=> ! [H] :
( in(H,G)
<=> ~ ( H != A
& H != B
& H != C
& H != D
& H != E
& H != F ) ) ) ).
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(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(A) ) ).
fof(rc5_funct_1,axiom,
? [A] :
( relation(A)
& relation_non_empty(A)
& function(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(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t6_ordinal1,conjecture,
! [A,B,C,D,E,F] :
~ ( in(A,B)
& in(B,C)
& in(C,D)
& in(D,E)
& in(E,F)
& in(F,A) ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t7_tarski,axiom,
! [A,B] :
~ ( in(A,B)
& ! [C] :
~ ( in(C,B)
& ! [D] :
~ ( in(D,B)
& in(D,C) ) ) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
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