TPTP Problem File: SET888+1.p
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%------------------------------------------------------------------------------
% File : SET888+1 : TPTP v9.0.0. Released v3.2.0.
% Domain : Set theory
% Problem : Basic properties of sets, theorem 29
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [Byl90] Bylinski (1990), Some Basic Properties of Sets
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : zfmisc_1__t29_zfmisc_1 [Urb06]
% Status : Theorem
% Rating : 0.70 v9.0.0, 0.72 v8.1.0, 0.69 v7.4.0, 0.63 v7.3.0, 0.59 v7.1.0, 0.65 v7.0.0, 0.73 v6.3.0, 0.75 v6.2.0, 0.92 v6.1.0, 0.93 v6.0.0, 0.91 v5.5.0, 0.89 v5.3.0, 0.93 v5.2.0, 0.95 v5.0.0, 0.96 v4.1.0, 0.91 v4.0.0, 0.88 v3.7.0, 0.90 v3.5.0, 0.89 v3.4.0, 0.95 v3.3.0, 0.86 v3.2.0
% Syntax : Number of formulae : 14 ( 7 unt; 0 def)
% Number of atoms : 25 ( 12 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 19 ( 8 ~; 1 |; 0 &)
% ( 6 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 5 ( 5 usr; 0 con; 1-2 aty)
% Number of variables : 30 ( 28 !; 2 ?)
% 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(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k5_xboole_0,axiom,
! [A,B] : symmetric_difference(A,B) = symmetric_difference(B,A) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d2_tarski,axiom,
! [A,B,C] :
( C = unordered_pair(A,B)
<=> ! [D] :
( in(D,C)
<=> ( D = A
| D = B ) ) ) ).
fof(d6_xboole_0,axiom,
! [A,B] : symmetric_difference(A,B) = set_union2(set_difference(A,B),set_difference(B,A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(t1_xboole_0,axiom,
! [A,B,C] :
( in(A,symmetric_difference(B,C))
<=> ~ ( in(A,B)
<=> in(A,C) ) ) ).
fof(t29_zfmisc_1,conjecture,
! [A,B] :
( A != B
=> symmetric_difference(singleton(A),singleton(B)) = unordered_pair(A,B) ) ).
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