## TPTP Problem File: SET914^7.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SET914^7 : TPTP v7.2.0. 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   : 0.78 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 (   0 unit;  42 type;  32 defn)
%            Number of atoms       :  566 (  36 equality; 265 variable)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :  435 (   5   ~;   5   |;   9   &; 406   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  192 ( 192   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   46 (  42   :;   0   =)
%            Number of variables   :  156 (   3 sgn;  41   !;   7   ?; 108   ^)
%                                         ( 156   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
%----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 ) ) ) ) ) ) )).

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