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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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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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