%------------------------------------------------------------------------------ % File : SET062^7 : TPTP v7.2.0. Released v5.5.0. % Domain : Set Theory % Problem : The empty set is a subset of all sets % Version : [Ben12] axioms. % English : % Refs : [Pas99] Pastre (1999), Email to G. Sutcliffe % : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe % Source : [Ben12] % Names : s4-cumul-SET062+4 [Ben12] % Status : Theorem % Rating : 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0 % Syntax : Number of formulae : 126 ( 0 unit; 48 type; 32 defn) % Number of atoms : 739 ( 36 equality; 338 variable) % Maximal formula depth : 14 ( 7 average) % Number of connectives : 594 ( 5 ~; 5 |; 9 &; 565 @) % ( 0 <=>; 10 =>; 0 <=; 0 <~>) % ( 0 ~|; 0 ~&) % Number of type conns : 201 ( 201 >; 0 *; 0 +; 0 <<) % Number of symbols : 52 ( 48 :; 0 =) % Number of variables : 196 ( 2 sgn; 55 !; 7 ?; 134 ^) % ( 196 :; 0 !>; 0 ?*) % ( 0 @-; 0 @+) % 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(equal_set_type,type,( equal_set: mu > mu > $i > $o )). thf(member_type,type,( member: mu > mu > $i > $o )). thf(subset_type,type,( subset: mu > mu > $i > $o )). thf(power_set_type,type,( power_set: mu > mu )). thf(existence_of_power_set_ax,axiom,( ! [V: $i,V1: mu] : ( exists_in_world @ ( power_set @ V1 ) @ V ) )). thf(intersection_type,type,( intersection: mu > mu > mu )). thf(existence_of_intersection_ax,axiom,( ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( intersection @ V2 @ V1 ) @ V ) )). thf(union_type,type,( union: mu > mu > mu )). thf(existence_of_union_ax,axiom,( ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( union @ V2 @ V1 ) @ V ) )). thf(difference_type,type,( difference: mu > mu > mu )). thf(existence_of_difference_ax,axiom,( ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( difference @ V2 @ V1 ) @ 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(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(sum_type,type,( sum: mu > mu )). thf(existence_of_sum_ax,axiom,( ! [V: $i,V1: mu] : ( exists_in_world @ ( sum @ V1 ) @ V ) )). thf(product_type,type,( product: mu > mu )). thf(existence_of_product_ax,axiom,( ! [V: $i,V1: mu] : ( exists_in_world @ ( product @ V1 ) @ V ) )). thf(empty_set_type,type,( empty_set: mu )). thf(existence_of_empty_set_ax,axiom,( ! [V: $i] : ( exists_in_world @ empty_set @ 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(difference_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ A @ C ) @ ( difference @ B @ C ) ) ) ) ) ) )). thf(difference_substitution_2,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ C @ A ) @ ( difference @ C @ B ) ) ) ) ) ) )). thf(intersection_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) )). thf(intersection_substitution_2,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) )). thf(power_set_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_set @ A ) @ ( power_set @ B ) ) ) ) ) )). thf(product_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( product @ A ) @ ( product @ B ) ) ) ) ) )). thf(singleton_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) )). thf(sum_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum @ A ) @ ( sum @ B ) ) ) ) ) )). thf(union_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) )). thf(union_substitution_2,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ 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(equal_set_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( equal_set @ A @ C ) ) @ ( equal_set @ B @ C ) ) ) ) ) )). thf(equal_set_substitution_2,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( equal_set @ C @ A ) ) @ ( equal_set @ C @ B ) ) ) ) ) )). thf(member_substitution_1,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) )). thf(member_substitution_2,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [C: mu] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ 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(subset,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mequiv @ ( subset @ A @ B ) @ ( mforall_ind @ ^ [X: mu] : ( mimplies @ ( member @ X @ A ) @ ( member @ X @ B ) ) ) ) ) ) )). thf(equal_set,axiom, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mequiv @ ( equal_set @ A @ B ) @ ( mand @ ( subset @ A @ B ) @ ( subset @ B @ A ) ) ) ) ) )). thf(power_set,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mequiv @ ( member @ X @ ( power_set @ A ) ) @ ( subset @ X @ A ) ) ) ) )). thf(intersection,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mequiv @ ( member @ X @ ( intersection @ A @ B ) ) @ ( mand @ ( member @ X @ A ) @ ( member @ X @ B ) ) ) ) ) ) )). thf(union,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mequiv @ ( member @ X @ ( union @ A @ B ) ) @ ( mor @ ( member @ X @ A ) @ ( member @ X @ B ) ) ) ) ) ) )). thf(empty_set,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mnot @ ( member @ X @ empty_set ) ) ) )). thf(difference,axiom, ( mvalid @ ( mforall_ind @ ^ [B: mu] : ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [E: mu] : ( mequiv @ ( member @ B @ ( difference @ E @ A ) ) @ ( mand @ ( member @ B @ E ) @ ( mnot @ ( member @ B @ A ) ) ) ) ) ) ) )). thf(singleton,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mequiv @ ( member @ X @ ( singleton @ A ) ) @ ( qmltpeq @ X @ A ) ) ) ) )). thf(unordered_pair,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mforall_ind @ ^ [B: mu] : ( mequiv @ ( member @ X @ ( unordered_pair @ A @ B ) ) @ ( mor @ ( qmltpeq @ X @ A ) @ ( qmltpeq @ X @ B ) ) ) ) ) ) )). thf(sum,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mequiv @ ( member @ X @ ( sum @ A ) ) @ ( mexists_ind @ ^ [Y: mu] : ( mand @ ( member @ Y @ A ) @ ( member @ X @ Y ) ) ) ) ) ) )). thf(product,axiom, ( mvalid @ ( mforall_ind @ ^ [X: mu] : ( mforall_ind @ ^ [A: mu] : ( mequiv @ ( member @ X @ ( product @ A ) ) @ ( mforall_ind @ ^ [Y: mu] : ( mimplies @ ( member @ Y @ A ) @ ( member @ X @ Y ) ) ) ) ) ) )). thf(thI15,conjecture, ( mvalid @ ( mforall_ind @ ^ [A: mu] : ( subset @ empty_set @ A ) ) )). 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