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Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-refrels 34401 Define the class of all reflexive relations. This is practically dfrefrels2 34403 (which reveals that RefRels can not include proper classes like I as is elements, cf. the comments of dfrefrels2 34403).

Another alternative definition is dfrefrels3 34404. The element of this class and the reflexive relation predicate (df-refrel 34402) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, cf. elrefrelsrel 34409.

This definition is similar to the definitions of the classes of all symmetric (df-symrels 34429) and transitive ( ~? df-trrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

RefRels = ( Refs ∩ Rels )
 
Definitiondf-refrel 34402 Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 34406. Alternate definitions are dfrefrel2 34405 and dfrefrel3 34406. The element of the class of all reflexive relations (df-refrels 34401) and this reflexive relation predicate are the same, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, cf. elrefrelsrel 34409. (Contributed by Peter Mazsa, 16-Jul-2021.)
( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfrefrels2 34403 Alternate definition of the class of all reflexive relations. This is a 0-ary class constant, which is recommended for definitions (cf. the 1. Guideline at http://us.metamath.org/ileuni/mathbox.html). Proper classes (like I, cf. iprc 7143) are not elements of this (or any) class: if a class is an element of another class, it is not a proper class but a set, cf. elex 3243. So if we use 0-ary constant classes as our main definitions, they are valid only for sets, not for proper classes. For proper classes we use predicate-type definitions like df-refrel 34402. Cf. the comment of df-rels 34375.

Note that while elementhood in the class of all relations cancels restriction of 𝑟 in dfrefrels2 34403, it keeps restriction of I: this is why the very similar definitions df-refs 34400, df-syms 34428 and ~? df-trs diverge when we switch from (general) sets to relations in dfrefrels2 34403, dfsymrels2 34431 and ~? dftrrels2 . (Contributed by Peter Mazsa, 20-Jul-2019.)

RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
 
Theoremdfrefrels3 34404* Alternate definition of the class of all reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.)
RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}
 
Theoremdfrefrel2 34405 Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
 
Theoremdfrefrel3 34406* Alternate definition of the reflexive relation predicate. A relation is reflexive iff: for all elements on its domain and range, if an element of its domain is the same as an element of its range, then there is the relation between them.

Note that this is definitely not the definition we are accustomed to, like e.g. issref 5544 or df-reflexive 42837 (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, cf. refsymrels3 34450. Only when this symmetry condition holds, like in case of equivalence relations, cf. ~? dfeqvrels3 , can we write the traditional form 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 for reflexive relations. For the special case with square Cartesian product when the two forms are equivalent cf. idinxpssinxp4 34232 where (∀𝑥𝐴𝑦𝐴(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑥𝑅𝑥). Cf. similar definition of the converse reflexive relations class dfcnvrefrel3 34419. (Contributed by Peter Mazsa, 8-Jul-2019.)

( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
 
Theoremelrefrels2 34407 Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
(𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
 
Theoremelrefrels3 34408* Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
(𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefrelsrel 34409 The element of the class of all reflexive relations and the reflexive relation predicate are the same, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
 
Theoremrefreleq 34410 Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
(𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
 
Theoremrefrelid 34411 Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
RefRel I
 
Theoremrefrelcoss 34412 The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
RefRel ≀ 𝑅
 
20.21.8  Converse reflexivity
 
Definitiondf-cnvrefs 34413 Define the class of all converse reflexive sets, cf. the comment of df-ssr 34388. It is used only by df-cnvrefrels 34414. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
 
Definitiondf-cnvrefrels 34414 Define the class of all converse reflexive relations. This is practically dfcnvrefrels2 34416 (which uses the traditional subclass relation ) : we use converse subset relation (brcnvssr 34396) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 21356), symmetric (df-syms 34428) and transitive ( ~? df-trs ) sets.

We use this concept to define functions ( ~? df-funsALTV , ~? df-funALTV ) and disjoints ( ~? df-disjs , ~? df-disjALTV ).

The element of the class of all converse reflexive relations and the converse reflexive relation predicate are the same, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, cf. elcnvrefrelsrel 34422. Alternate definitions are dfcnvrefrels2 34416 and dfcnvrefrels3 34417. (Contributed by Peter Mazsa, 7-Jul-2019.)

CnvRefRels = ( CnvRefs ∩ Rels )
 
Definitiondf-cnvrefrel 34415 Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), cf. the comment of dfcnvrefrel3 34419. Alternate definitions are dfcnvrefrel2 34418 and dfcnvrefrel3 34419. (Contributed by Peter Mazsa, 16-Jul-2021.)
( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfcnvrefrels2 34416 Alternate definition of the class of all converse reflexive relations. Cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 21-Jul-2021.)
CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
 
Theoremdfcnvrefrels3 34417* Alternate definition of the class of all converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
 
Theoremdfcnvrefrel2 34418 Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019.)
( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfcnvrefrel3 34419* Alternate definition of the converse reflexive relation predicate. A relation is converse reflexive iff: for all elements on its domain and range, if for an element of its domain and for an element of its range there is the relation between them, then the two elements are the same, cf. the comment of dfrefrel3 34406. (Contributed by Peter Mazsa, 25-Jul-2021.)
( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
 
Theoremelcnvrefrels2 34420 Element of the class of all converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.)
(𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
 
Theoremelcnvrefrels3 34421* Element of the class of all converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.)
(𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))
 
Theoremelcnvrefrelsrel 34422 The element of the class of all converse reflexive relations and the converse reflexive relation predicate are the same, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
 
Theoremcnvrefrelcoss2 34423 Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.)
( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
 
Theoremcosselcnvrefrels2 34424 Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.)
( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))
 
Theoremcosselcnvrefrels3 34425* Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021.)
( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
 
Theoremcosselcnvrefrels4 34426* Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021.)
( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels ))
 
Theoremcosselcnvrefrels5 34427* Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021.)
( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
 
20.21.9  Symmetry
 
Definitiondf-syms 34428 Define the class of all symmetric sets. It is used only by df-symrels 34429.

Note the similarity of the definitions df-refs 34400, df-syms 34428 and ~? df-trs , cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 19-Jul-2019.)

Syms = {𝑥(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
 
Definitiondf-symrels 34429 Define the class of all symmetric relations. The element of the class of all symmetric relations and the symmetric relation predicate are the same when 𝑅 is a set, cf. elsymrelsrel 34443. Alternate definitions are dfsymrels2 34431, dfsymrels3 34432, dfsymrels4 34433 and dfsymrels5 34434.

This definition is similar to the definitions of the classes of all reflexive (df-refrels 34401) and transitive ( ~? df-trrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

SymRels = ( Syms ∩ Rels )
 
Definitiondf-symrel 34430 Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) The element of the class of all symmetric relations and the symmetric relation predicate are the same when 𝑅 is a set, cf. elsymrelsrel 34443. Alternate definitions are dfsymrel2 34435 and dfsymrel3 34436. (Contributed by Peter Mazsa, 16-Jul-2021.)
( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfsymrels2 34431 Alternate definition of the class of all symmetric relations. Cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
 
Theoremdfsymrels3 34432* Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
 
Theoremdfsymrels4 34433 Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟 = 𝑟}
 
Theoremdfsymrels5 34434* Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
 
Theoremdfsymrel2 34435 Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
 
Theoremdfsymrel3 34436* Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
 
Theoremdfsymrel4 34437 Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
 
Theoremdfsymrel5 34438* Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
 
Theoremelsymrels2 34439 Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
 
Theoremelsymrels3 34440* Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
 
Theoremelsymrels4 34441 Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅 = 𝑅𝑅 ∈ Rels ))
 
Theoremelsymrels5 34442* Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
 
Theoremelsymrelsrel 34443 The element of the class of all symmetric relations and the symmetric relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
 
Theoremsymreleq 34444 Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
(𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
 
Theoremsymrelim 34445 Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
 
Theoremsymrelcoss 34446 The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
SymRel ≀ 𝑅
 
20.21.10  Reflexivity and symmetry
 
Theoremsymrefref2 34447 Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, cf. symrefref3 34448. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
 
Theoremsymrefref3 34448* Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, cf. symrefref2 34447. (Contributed by Peter Mazsa, 23-Aug-2021.)
(∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
 
Theoremrefsymrels2 34449 Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~? dfeqvrels2 ) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 34403, cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 20-Jul-2019.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
 
Theoremrefsymrels3 34450* Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~? dfeqvrels3 ) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 34404, cf. the comment of dfrefrels3 34404. (Contributed by Peter Mazsa, 22-Jul-2019.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
 
Theoremrefsymrel2 34451 A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 34405, cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
 
Theoremrefsymrel3 34452* A relation which is reflexive and symmetric (like an equivalence relation) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for its reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrel3 34406, cf. the comment of dfrefrel3 34406. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
 
Theoremelrefsymrels2 34453 Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~? dfeqvrels2 ) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 34403, cf. the comment of dfrefrels2 34403. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrels3 34454* Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~? dfeqvrels3 ) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrels3 34404, cf. the comment of dfrefrels3 34404. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrelsrel 34455 The element of the class of all reflexive and symmetric relations and the conjunction of the reflexive and symmetric relation predicates are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 23-Aug-2021.)
(𝑅𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
 
20.22  Mathbox for Rodolfo Medina
 
20.22.1  Partitions
 
Theoremprtlem60 34456 Lemma for prter3 34486. (Contributed by Rodolfo Medina, 9-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))
 
Theorembicomdd 34457 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjca2r 34458 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjca3 34459 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))
 
Theoremprtlem70 34460 Lemma for prter3 34486: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
((((𝜓𝜂) ∧ ((𝜑𝜃) ∧ (𝜒𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃𝜏)))) ∧ 𝜂))
 
Theoremibdr 34461 Reverse of ibd 258. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(𝜑 → (𝜒 → (𝜓𝜒)))       (𝜑 → (𝜒𝜓))
 
Theorempm5.31r 34462 Variant of pm5.31 611. (Contributed by Rodolfo Medina, 15-Oct-2010.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜒𝜓)))
 
Theoremprtlem100 34463 Lemma for prter3 34486. (Contributed by Rodolfo Medina, 19-Oct-2010.)
(∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
 
Theoremprtlem5 34464* Lemma for prter1 34483, prter2 34485, prter3 34486 and prtex 34484. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
 
Theoremprtlem80 34465 Lemma for prter2 34485. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
 
Theorembrabsb2 34466* A closed form of brabsb 5015. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
 
Theoremeqbrrdv2 34467* Other version of eqbrrdiv 5252. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremprtlem9 34468* Lemma for prter3 34486. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
 
Theoremprtlem10 34469* Lemma for prter3 34486. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))
 
Theoremprtlem11 34470 Lemma for prter2 34485. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
 
Theoremprtlem12 34471* Lemma for prtex 34484 and prter3 34486. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )
 
Theoremprtlem13 34472* Lemma for prter1 34483, prter2 34485, prter3 34486 and prtex 34484. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
 
Theoremprtlem16 34473* Lemma for prtex 34484, prter2 34485 and prter3 34486. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       dom = 𝐴
 
Theoremprtlem400 34474* Lemma for prter2 34485 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}        ¬ ∅ ∈ ( 𝐴 / )
 
Syntaxwprt 34475 Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴
 
Definitiondf-prt 34476* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
 
Theoremerprt 34477 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( Er 𝑋 → Prt (𝐴 / ))
 
Theoremprtlem14 34478* Lemma for prter1 34483, prter2 34485 and prtex 34484. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
 
Theoremprtlem15 34479* Lemma for prter1 34483 and prtex 34484. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))
 
Theoremprtlem17 34480* Lemma for prter2 34485. (Contributed by Rodolfo Medina, 15-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
 
Theoremprtlem18 34481* Lemma for prter2 34485. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
 
Theoremprtlem19 34482* Lemma for prter2 34485. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))
 
Theoremprter1 34483* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 Er 𝐴)
 
Theoremprtex 34484* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
 
Theoremprter2 34485* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))
 
Theoremprter3 34486* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)
 
20.23  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 34590, means that the definition or theorem is not used for the derivation of hlathil 37570. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 37570.

 
20.23.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16

These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2091, axc7 2170, axc10 2288, axc11 2347, axc11n 2342, axc15 2339, axc9 2338, axc14 2400, and axc16 2173.

 
Axiomax-c5 34487 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all 𝑥, it is true for any specific 𝑥 (that would typically occur as a free variable in the wff substituted for 𝜑). (A free variable is one that does not occur in the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦, but only 𝑥 is free in 𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1762. Conditional forms of the converse are given by ax-13 2282, ax-c14 34495, ax-c16 34496, and ax-5 1879.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2381.

An interesting alternate axiomatization uses axc5c711 34522 and ax-c4 34488 in place of ax-c5 34487, ax-4 1777, ax-10 2059, and ax-11 2074.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2091. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥𝜑𝜑)
 
Axiomax-c4 34488 Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying 𝜓. Notice that 𝑥 must not be a free variable in the antecedent of the quantified implication, and we express this by binding 𝜑 to "protect" the axiom from a 𝜑 containing a free 𝑥. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc4 2168. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Axiomax-c7 34489 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use axc5c711 34522 in place of ax-c5 34487, ax-c7 34489, and ax-11 2074.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 2170. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
 
Axiomax-c10 34490 A variant of ax6 2287. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem axc10 2288. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Axiomax-c11 34491 Axiom ax-c11 34491 was the original version of ax-c11n 34492 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 34492 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11 2347. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
Axiomax-c11n 34492 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-c11 34491 and was replaced with this shorter ax-c11n 34492 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2347. Conversely, this axiom is proved from ax-c11 34491 as theorem axc11nfromc11 34530.

This axiom was proved redundant in July 2015. See theorem axc11n 2342.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2342. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Axiomax-c15 34493 Axiom ax-c15 34493 was the original version of ax-12 2087, before it was discovered (in Jan. 2007) that the shorter ax-12 2087 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."

Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-c15 34493 (from which the ax-12 2087 instance follows by theorem ax12 2340.) The proof is by induction on formula length, using ax12eq 34545 and ax12el 34546 for the basis steps and ax12indn 34547, ax12indi 34548, and ax12inda 34552 for the induction steps. (This paragraph is true provided we use ax-c11 34491 in place of ax-c11n 34492.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2339, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 
Axiomax-c9 34494 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2338. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
Axiomax-c14 34495 Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-5 1879; see theorem axc14 2400. Alternately, ax-5 1879 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1879. We retain ax-c14 34495 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1879, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc14 2400. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
 
Axiomax-c16 34496* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1879 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 4887), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1879; see theorem axc16 2173. Alternately, ax-5 1879 becomes logically redundant in the presence of this axiom, but without ax-5 1879 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 34496 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1879, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc16 2173. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old

Theorems ax12fromc15 34509 and ax13fromc9 34510 require some intermediate theorems that are included in this section.

 
Theoremaxc5 34497 This theorem repeats sp 2091 under the name axc5 34497, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-c5 34487. It is preferred that references to this theorem use the name sp 2091. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremax4fromc4 34498 Rederivation of axiom ax-4 1777 from ax-c4 34488, ax-c5 34487, ax-gen 1762 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2168 for the derivation of ax-c4 34488 from ax-4 1777. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremax10fromc7 34499 Rederivation of axiom ax-10 2059 from ax-c7 34489, ax-c4 34488, ax-c5 34487, ax-gen 1762 and propositional calculus. See axc7 2170 for the derivation of ax-c7 34489 from ax-10 2059. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremax6fromc10 34500 Rederivation of axiom ax-6 1945 from ax-c7 34489, ax-c10 34490, ax-gen 1762 and propositional calculus. See axc10 2288 for the derivation of ax-c10 34490 from ax-6 1945. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦
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