TPTP Problem File: SET908^7.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SET908^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : Set Theory
% Problem : union(singleton(A),B) != empty_set
% Version : [Ben12] axioms.
% English :
% Refs : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
% : [Byl90] Bylinski (1990), Some Basic Properties of Sets
% : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names : s4-cumul-GSE908+1 [Ben12]
% Status : Theorem
% Rating : 0.90 v8.2.0, 0.92 v8.1.0, 1.00 v5.5.0
% Syntax : Number of formulae : 101 ( 36 unt; 41 typ; 32 def)
% Number of atoms : 429 ( 36 equ; 0 cnn)
% Maximal formula atoms : 58 ( 7 avg)
% Number of connectives : 596 ( 5 ~; 5 |; 9 &; 567 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 188 ( 188 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 50 usr; 12 con; 0-3 aty)
% Number of variables : 144 ( 97 ^; 40 !; 7 ?; 144 :)
% SPC : TH0_THM_EQU_NAR
% Comments : Goedel translation of SET908+1
%------------------------------------------------------------------------------
%----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(in_type,type,
in: mu > mu > $i > $o ).
thf(empty_type,type,
empty: 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(singleton_type,type,
singleton: mu > mu ).
thf(existence_of_singleton_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( singleton @ V1 ) @ V ) ).
thf(set_union2_type,type,
set_union2: mu > mu > mu ).
thf(existence_of_set_union2_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( set_union2 @ V2 @ V1 ) @ V ) ).
thf(reflexivity,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] : ( mbox_s4 @ ( qmltpeq @ X @ X ) ) ) ) ) ).
thf(symmetry,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ X ) ) ) ) ) ) ) ) ) ).
thf(transitivity,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Z: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ Z ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
thf(set_union2_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( set_union2 @ A @ C ) @ ( set_union2 @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(set_union2_substitution_2,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( set_union2 @ C @ A ) @ ( set_union2 @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(singleton_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ) ) ) ) ) ).
thf(empty_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( empty @ A ) ) ) @ ( mbox_s4 @ ( empty @ B ) ) ) ) ) ) ) ) ) ).
thf(in_substitution_1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( in @ A @ C ) ) ) @ ( mbox_s4 @ ( in @ B @ C ) ) ) ) ) ) ) ) ) ) ) ).
thf(in_substitution_2,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( in @ C @ A ) ) ) @ ( mbox_s4 @ ( in @ C @ B ) ) ) ) ) ) ) ) ) ) ) ).
thf(antisymmetry_r2_hidden,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( in @ A @ B ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( in @ B @ A ) ) ) ) ) ) ) ) ) ) ) ).
thf(commutativity_k2_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( qmltpeq @ ( set_union2 @ A @ B ) @ ( set_union2 @ B @ A ) ) ) ) ) ) ) ) ).
thf(d1_tarski,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mand
@ ( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ B @ ( singleton @ A ) ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( in @ C @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ C @ A ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ C @ A ) ) @ ( mbox_s4 @ ( in @ C @ B ) ) ) ) ) ) ) ) )
@ ( mbox_s4
@ ( mimplies
@ ( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] : ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( in @ C @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ C @ A ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ C @ A ) ) @ ( mbox_s4 @ ( in @ C @ B ) ) ) ) ) ) )
@ ( mbox_s4 @ ( qmltpeq @ B @ ( singleton @ A ) ) ) ) ) ) ) ) ) ) ) ).
thf(d1_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mand
@ ( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ empty_set ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( in @ B @ A ) ) ) ) ) ) ) )
@ ( mbox_s4
@ ( mimplies
@ ( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( in @ B @ A ) ) ) ) ) )
@ ( mbox_s4 @ ( qmltpeq @ A @ empty_set ) ) ) ) ) ) ) ) ).
thf(d2_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [C: mu] :
( mand
@ ( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ C @ ( set_union2 @ A @ B ) ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [D: mu] : ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( in @ D @ C ) ) @ ( mor @ ( mbox_s4 @ ( in @ D @ A ) ) @ ( mbox_s4 @ ( in @ D @ B ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mor @ ( mbox_s4 @ ( in @ D @ A ) ) @ ( mbox_s4 @ ( in @ D @ B ) ) ) @ ( mbox_s4 @ ( in @ D @ C ) ) ) ) ) ) ) ) )
@ ( mbox_s4
@ ( mimplies
@ ( mbox_s4
@ ( mforall_ind
@ ^ [D: mu] : ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( in @ D @ C ) ) @ ( mor @ ( mbox_s4 @ ( in @ D @ A ) ) @ ( mbox_s4 @ ( in @ D @ B ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mor @ ( mbox_s4 @ ( in @ D @ A ) ) @ ( mbox_s4 @ ( in @ D @ B ) ) ) @ ( mbox_s4 @ ( in @ D @ C ) ) ) ) ) ) )
@ ( mbox_s4 @ ( qmltpeq @ C @ ( set_union2 @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(fc1_xboole_0,axiom,
mvalid @ ( mbox_s4 @ ( empty @ empty_set ) ) ).
thf(fc2_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( empty @ A ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( empty @ ( set_union2 @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(fc3_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( empty @ A ) ) ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( empty @ ( set_union2 @ B @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(idempotence_k2_xboole_0,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( qmltpeq @ ( set_union2 @ A @ A ) @ A ) ) ) ) ) ) ) ).
thf(rc1_xboole_0,axiom,
( mvalid
@ ( mexists_ind
@ ^ [A: mu] : ( mbox_s4 @ ( empty @ A ) ) ) ) ).
thf(rc2_xboole_0,axiom,
( mvalid
@ ( mexists_ind
@ ^ [A: mu] : ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( empty @ A ) ) ) ) ) ) ).
thf(t49_zfmisc_1,conjecture,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [A: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [B: mu] : ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( qmltpeq @ ( set_union2 @ ( singleton @ A ) @ B ) @ empty_set ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------