TPTP Problem File: SET914^7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET914^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : Set Theory
% Problem : ~ ( disjoint(unordered_pair(A,B),C) & in(A,C) )
% Version : [Ben12] axioms.
% English :
% Refs : [Byl90] Bylinski (1990), Some Basic Properties of Sets
% : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names : s4-cumul-SET914+1 [Ben12]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.62 v8.1.0, 0.45 v7.5.0, 0.29 v7.4.0, 0.67 v7.3.0, 0.78 v7.2.0, 0.75 v7.1.0, 0.88 v7.0.0, 0.86 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.86 v5.5.0
% Syntax : Number of formulae : 106 ( 36 unt; 42 typ; 32 def)
% Number of atoms : 278 ( 36 equ; 0 cnn)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 435 ( 5 ~; 5 |; 9 &; 406 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 192 ( 192 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 51 usr; 12 con; 0-3 aty)
% Number of variables : 156 ( 108 ^; 41 !; 7 ?; 156 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
%------------------------------------------------------------------------------
%----Include axioms for Modal logic S4 under cumulative domains
include('Axioms/LCL015^0.ax').
include('Axioms/LCL013^5.ax').
include('Axioms/LCL015^1.ax').
%------------------------------------------------------------------------------
thf(empty_type,type,
empty: mu > $i > $o ).
thf(in_type,type,
in: mu > mu > $i > $o ).
thf(disjoint_type,type,
disjoint: mu > mu > $i > $o ).
thf(empty_set_type,type,
empty_set: mu ).
thf(existence_of_empty_set_ax,axiom,
! [V: $i] : ( exists_in_world @ empty_set @ V ) ).
thf(set_intersection2_type,type,
set_intersection2: mu > mu > mu ).
thf(existence_of_set_intersection2_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( set_intersection2 @ V2 @ V1 ) @ V ) ).
thf(unordered_pair_type,type,
unordered_pair: mu > mu > mu ).
thf(existence_of_unordered_pair_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( unordered_pair @ V2 @ V1 ) @ V ) ).
thf(reflexivity,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] : ( qmltpeq @ X @ X ) ) ) ).
thf(symmetry,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
thf(transitivity,axiom,
( mvalid
@ ( mforall_ind
@ ^ [X: mu] :
( mforall_ind
@ ^ [Y: mu] :
( mforall_ind
@ ^ [Z: mu] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
thf(set_intersection2_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_intersection2 @ A @ C ) @ ( set_intersection2 @ B @ C ) ) ) ) ) ) ) ).
thf(set_intersection2_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_intersection2 @ C @ A ) @ ( set_intersection2 @ C @ B ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ) ).
thf(unordered_pair_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ) ).
thf(disjoint_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ A @ C ) ) @ ( disjoint @ B @ C ) ) ) ) ) ) ).
thf(disjoint_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ C @ A ) ) @ ( disjoint @ C @ B ) ) ) ) ) ) ).
thf(empty_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( empty @ A ) ) @ ( empty @ B ) ) ) ) ) ).
thf(in_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( in @ A @ C ) ) @ ( in @ B @ C ) ) ) ) ) ) ).
thf(in_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( in @ C @ A ) ) @ ( in @ C @ B ) ) ) ) ) ) ).
thf(antisymmetry_r2_hidden,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( in @ A @ B ) @ ( mnot @ ( in @ B @ A ) ) ) ) ) ) ).
thf(commutativity_k2_tarski,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( qmltpeq @ ( unordered_pair @ A @ B ) @ ( unordered_pair @ B @ A ) ) ) ) ) ).
thf(commutativity_k3_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( qmltpeq @ ( set_intersection2 @ A @ B ) @ ( set_intersection2 @ B @ A ) ) ) ) ) ).
thf(d1_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mequiv @ ( qmltpeq @ A @ empty_set )
@ ( mforall_ind
@ ^ [B: mu] : ( mnot @ ( in @ B @ A ) ) ) ) ) ) ).
thf(d2_tarski,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] :
( mequiv @ ( qmltpeq @ C @ ( unordered_pair @ A @ B ) )
@ ( mforall_ind
@ ^ [D: mu] : ( mequiv @ ( in @ D @ C ) @ ( mor @ ( qmltpeq @ D @ A ) @ ( qmltpeq @ D @ B ) ) ) ) ) ) ) ) ) ).
thf(d3_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] :
( mequiv @ ( qmltpeq @ C @ ( set_intersection2 @ A @ B ) )
@ ( mforall_ind
@ ^ [D: mu] : ( mequiv @ ( in @ D @ C ) @ ( mand @ ( in @ D @ A ) @ ( in @ D @ B ) ) ) ) ) ) ) ) ) ).
thf(d7_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mequiv @ ( disjoint @ A @ B ) @ ( qmltpeq @ ( set_intersection2 @ A @ B ) @ empty_set ) ) ) ) ) ).
thf(fc1_xboole_0,axiom,
mvalid @ ( empty @ empty_set ) ).
thf(idempotence_k3_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( qmltpeq @ ( set_intersection2 @ A @ A ) @ A ) ) ) ) ).
thf(rc1_xboole_0,axiom,
( mvalid
@ ( mexists_ind
@ ^ [A: mu] : ( empty @ A ) ) ) ).
thf(rc2_xboole_0,axiom,
( mvalid
@ ( mexists_ind
@ ^ [A: mu] : ( mnot @ ( empty @ A ) ) ) ) ).
thf(symmetry_r1_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( disjoint @ A @ B ) @ ( disjoint @ B @ A ) ) ) ) ) ).
thf(t55_zfmisc_1,conjecture,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mnot @ ( mand @ ( disjoint @ ( unordered_pair @ A @ B ) @ C ) @ ( in @ A @ C ) ) ) ) ) ) ) ).
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