TPTP Problem File: SET907^7.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SET907^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : Set Theory
% Problem : ( in(A,B) & in(C,B) ) => set_union2(unordered_pair(A,C),B) = B
% 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-SET907+1 [Ben12]
% Status : Theorem
% Rating : 0.10 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v5.5.0
% Syntax : Number of formulae : 103 ( 35 unt; 41 typ; 32 def)
% Number of atoms : 268 ( 36 equ; 0 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 418 ( 5 ~; 5 |; 9 &; 389 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 192 ( 192 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 49 usr; 10 con; 0-3 aty)
% Number of variables : 152 ( 105 ^; 40 !; 7 ?; 152 :)
% 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(subset_type,type,
subset: mu > mu > $i > $o ).
thf(in_type,type,
in: mu > mu > $i > $o ).
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(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
@ ( 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_union2_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_union2 @ A @ C ) @ ( set_union2 @ B @ C ) ) ) ) ) ) ) ).
thf(set_union2_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_union2 @ C @ A ) @ ( set_union2 @ 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(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(subset_substitution_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ A @ C ) ) @ ( subset @ B @ C ) ) ) ) ) ) ).
thf(subset_substitution_2,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ C @ A ) ) @ ( subset @ 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_k2_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( qmltpeq @ ( set_union2 @ A @ B ) @ ( set_union2 @ B @ A ) ) ) ) ) ).
thf(fc2_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( mnot @ ( empty @ A ) ) @ ( mnot @ ( empty @ ( set_union2 @ A @ B ) ) ) ) ) ) ) ).
thf(fc3_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( mnot @ ( empty @ A ) ) @ ( mnot @ ( empty @ ( set_union2 @ B @ A ) ) ) ) ) ) ) ).
thf(idempotence_k2_xboole_0,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( qmltpeq @ ( set_union2 @ 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(reflexivity_r1_tarski,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( subset @ A @ A ) ) ) ) ).
thf(t12_xboole_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] : ( mimplies @ ( subset @ A @ B ) @ ( qmltpeq @ ( set_union2 @ A @ B ) @ B ) ) ) ) ) ).
thf(t38_zfmisc_1,axiom,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mequiv @ ( subset @ ( unordered_pair @ A @ B ) @ C ) @ ( mand @ ( in @ A @ C ) @ ( in @ B @ C ) ) ) ) ) ) ) ).
thf(t48_zfmisc_1,conjecture,
( mvalid
@ ( mforall_ind
@ ^ [A: mu] :
( mforall_ind
@ ^ [B: mu] :
( mforall_ind
@ ^ [C: mu] : ( mimplies @ ( mand @ ( in @ A @ B ) @ ( in @ C @ B ) ) @ ( qmltpeq @ ( set_union2 @ ( unordered_pair @ A @ C ) @ B ) @ B ) ) ) ) ) ) ).
%------------------------------------------------------------------------------