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Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempfxco 40301 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝐴𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹𝑊) prefix 𝑁))

21.34.7  Auxiliary theorems for graph theory

Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

21.34.7.1  Negated equality and membership - extension

Theoremelnelall 40302 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴𝐵 → (𝐴𝐵𝜑))

21.34.7.2  Subclasses and subsets - extension

Theoremclel5 40303* Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.)
(𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)

Theoremdfss7 40304* Alternate definition of subclass relationship: a class 𝐴 is a subclass of another class 𝐵 iff each element of 𝐴 is equal to an element of 𝐵. (Contributed by AV, 13-Nov-2020.)
(𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)

Theoremsssseq 40305 If a class is a subclass of another class, the classes are equal iff the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)
(𝐵𝐴 → (𝐴𝐵𝐴 = 𝐵))

Theoremprcssprc 40306 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

21.34.7.3  The empty set - extension

Theoremralnralall 40307* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))

Theoremfalseral0 40308* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)

Theoremralralimp 40309* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))

Theoremn0rex 40310* There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)

Theoremssn0rex 40311* There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)

21.34.7.4  Unordered and ordered pairs - extension

Theoremelpwdifsn 40312 A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)
((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))

Theorempr1eqbg 40313 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Theorempr1nebg 40314 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶}))

Theoremrexdifpr 40315 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))

Theoremopidg 40316 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4359. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 18-Sep-2021.)
(𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

21.34.7.5  Indexed union and intersection - extension

TheoremotiunsndisjX 40317* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
(𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})

21.34.7.6  Ordered-pair class abstractions - extension

Theoremopabn1stprc 40318* An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wwf. (Contributed by AV, 27-Dec-2020.)
(∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

21.34.7.7  Relations - extension

Theoremresresdm 40319 A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
(𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Theoremresisresindm 40320 The restriction of a relation by a set 𝐵 is identical with the restriction by the intersection of 𝐵 with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
(Rel 𝐹 → (𝐹𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹)))

21.34.7.8  Functions - extension

Theoremfvifeq 40321 Equality of function values with conditional arguments, see also fvif 6114. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))

Theorem2f1fvneq 40322 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(((𝐸:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))

Theoremf1cofveqaeq 40323 If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
(((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Theoremf1cofveqaeqALT 40324 Alternate proof of f1cofveqaeq 40323, 1 essential step shorter, but having more bytes (305 vs. 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Theoremrnfdmpr 40325 The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))

Theoremimarnf1pr 40326 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function of a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))

Theoremfunop1 40327* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
(∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Theoremf1ssf1 40328 A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)

Theoremfun2dmnopgexmpl 40329 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))

Theoremopabresex0d 40330* A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfex0d 40331* A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Theoremopabresexd 40332* A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Theoremopabbrfexd 40333* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Theoremopabresex2d 40334* Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)

Theoremmptmpt2opabbrd 40335* The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.)
(𝜑𝐺𝑊)    &   (𝜑𝑋 ∈ (𝐴𝐺))    &   (𝜑𝑌 ∈ (𝐵𝐺))    &   (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)    &   ((𝜑𝑓(𝐷𝐺)) → 𝜓)    &   ((𝑎 = 𝑋𝑏 = 𝑌) → (𝜏𝜃))    &   (𝑔 = 𝐺 → (𝜒𝜏))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝜒𝑓(𝐷𝑔))}))       (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝜃𝑓(𝐷𝐺))})

Theoremmptmpt2opabovd 40336* The operation value of a function value of a collection of ordered pairs of related elements (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.)
(𝜑𝐺𝑊)    &   (𝜑𝑋 ∈ (𝐴𝐺))    &   (𝜑𝑌 ∈ (𝐵𝐺))    &   (𝜑 → {⟨𝑓, ⟩ ∣ 𝜓} ∈ 𝑉)    &   ((𝜑𝑓(𝐷𝐺)) → 𝜓)    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴𝑔), 𝑏 ∈ (𝐵𝑔) ↦ {⟨𝑓, ⟩ ∣ (𝑓(𝑎(𝐶𝑔)𝑏)𝑓(𝐷𝑔))}))       (𝜑 → (𝑋(𝑀𝐺)𝑌) = {⟨𝑓, ⟩ ∣ (𝑓(𝑋(𝐶𝐺)𝑌)𝑓(𝐷𝐺))})

Theoremfpropnf1 40337 A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩}       (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹 ∧ ¬ Fun 𝐹))

21.34.7.9  Restricted iota - extension

Theoremriotaeqimp 40338* If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
𝐼 = (𝑎𝑉 𝑋 = 𝐴)    &   𝐽 = (𝑎𝑉 𝑌 = 𝐴)    &   (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)    &   (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)       ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)

21.34.7.10  Equinumerosity - extension

Theoremresfnfinfin 40339 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)

Theoremresidfi 40340 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
(( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

21.34.7.11  Subtraction - extension

Theoremcnambpcma 40341 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) − 𝐴) = (𝐶𝐵))

Theoremcnapbmcpd 40342 ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) − 𝐶) + 𝐷) = (((𝐴 + 𝐷) + 𝐵) − 𝐶))

21.34.7.12  Ordering on reals (cont.) - extension

Theoremleaddsuble 40343 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))

Theorem2leaddle2 40344 If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶)))

Theoremltnltne 40345 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴)))

Theoremp1lep2 40346 A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ ℝ → (𝑁 + 1) ≤ (𝑁 + 2))

Theoremltsubsubaddltsub 40347 If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝐽 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → (𝐽 < ((𝐿𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿𝑁)))

Theoremzm1nn 40348 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝑁 ∈ ℕ0𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽𝐽 < ((𝐿𝑁) − 1)) → (𝐿 − 1) ∈ ℕ))

21.34.7.13  Nonnegative integers (as a subset of complex numbers) - extension

Theoremnn0resubcl 40349 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ ℝ)

21.34.7.14  Upper sets of integers - extension

Theoremeluzge0nn0 40350 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
(𝑁 ∈ (ℤ𝑀) → (0 ≤ 𝑀𝑁 ∈ ℕ0))

21.34.7.15  Finite intervals of integers - extension

Theoremssfz12 40351 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀𝐾𝐿𝑁)))

Theoremelfz2z 40352 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ (0 ≤ 𝐾𝐾𝑁)))

Theorem2elfz3nn0 40353 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0))

Theoremfz0addcom 40354 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theorem2elfz2melfz 40355 If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁𝐴)) ∈ (0...𝐴)))

Theoremfz0addge0 40356 The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ (0...𝑀) ∧ 𝐵 ∈ (0...𝑁)) → 0 ≤ (𝐴 + 𝐵))

Theoremelfzlble 40357 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ((𝑁𝑀)...𝑁))

Theoremelfzelfzlble 40358 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑁 < (𝑀 + 𝐾)) → 𝐾 ∈ ((𝑁𝑀)...𝑁))

21.34.7.16  Half-open integer ranges - extension

Theoremsubsubelfzo0 40359 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐼 ∈ (0..^𝑁) ∧ ¬ 𝐼 < (𝑁𝐴)) → (𝐼 − (𝑁𝐴)) ∈ (0..^𝐴))

Theoremfzoopth 40360 A half-open integer range can represent an ordered pair, analogous to fzopth 12249. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

Theorem2ffzoeq 40361* Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))

Theoremfzosplitpr 40362 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(𝐵 ∈ (ℤ𝐴) → (𝐴..^(𝐵 + 2)) = ((𝐴..^𝐵) ∪ {𝐵, (𝐵 + 1)}))

Theoremprinfzo0 40363 The intersection of a half-open integer range and the pair of its outer borders is empty. (Contributed by AV, 9-Jan-2021.)
(𝑀 ∈ ℤ → ({𝑀, 𝑁} ∩ ((𝑀 + 1)..^𝑁)) = ∅)

Theoremelfzr 40364 A member of a finite interval of integers is either a member of the corresponding half-open integer range or the upper bound of the interval. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾 ∈ (𝑀..^𝑁) ∨ 𝐾 = 𝑁))

Theoremelfzo0l 40365 A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (0..^𝑁) → (𝐾 = 0 ∨ 𝐾 ∈ (1..^𝑁)))

Theoremelfzlmr 40366 A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))

Theoremelfz0lmr 40367 A member of a finite interval of nonnegative integers is either 0 or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.)
(𝐾 ∈ (0...𝑁) → (𝐾 = 0 ∨ 𝐾 ∈ (1..^𝑁) ∨ 𝐾 = 𝑁))

Theoremresunimafz0 40368 Formerly part of proof of eupth2lem3 41404: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
(𝜑 → Fun 𝐼)    &   (𝜑𝐹:(0..^(#‘𝐹))⟶dom 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))       (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))

21.34.7.17  The ` # ` (set size) function - extension

Theoremnfile 40369 The size of any infinite set is always greater than or equal to the the size of any set. (Contributed by AV, 13-Nov-2020.)
((𝐴𝑉𝐵𝑊 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵))

Theoremhash1n0 40370 If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
((𝐴𝑉 ∧ (#‘𝐴) = 1) → 𝐴 ≠ ∅)

21.34.7.18  Finite and infinite sums - extension

Theoremfsummsndifre 40371* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ)

Theoremfsumsplitsndif 40372* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑋𝐴 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + 𝑋 / 𝑘𝐵))

Theoremfsummmodsndifre 40373* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ)

Theoremfsummmodsnunz 40374* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ)

21.34.8  Graph theory (revised, deprecated)

21.34.8.1  Undirected hypergraphs (deprecated)

Theoremuhgruhgra 40375 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝐺 ∈ UHGraph ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝑉 UHGrph 𝐸)

Theoremuhgrauhgr 40376 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝑉 UHGrph 𝐸𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐺𝑊𝐺 ∈ UHGraph ))

Theoremuhgrauhgrbi 40377 Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
((𝐺𝑊𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝑉 UHGrph 𝐸𝐺 ∈ UHGraph ))

21.34.8.2  Undirected simple graphs - basics

For undirected graphs, we will have the following hierarchy/taxonomy:

* Undirected Hypergraph: UHGraph

* Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 25735)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 25768)

* Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph

* Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ULFHGraph

* Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 40406) and USPGraph ⊆ USHGraph (uspgrushgr 40405), see also uspgrupgrushgr 40407

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 25769) and UMGraph ⊆ ULFHGraph (umgrislfupgr 25789)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 40408) and USGraph ⊆ UMGraph (usgrumgr 40409) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 40414) see also usgrumgruspgr 40410

In this section, "simple graph" will always stand for "undirected simple graph (without loops)" and "simple pseudograph" for "undirected simple pseudograph (which could have loops)".

Syntaxcuspgr 40378 Extend class notation with undirected simple pseudographs (which could have loops).
class USPGraph

Syntaxcusgr 40379 Extend class notation with undirected simple graphs (without loops).
class USGraph

Definitiondf-uspgr 40380* Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 40406) or a special undirected simple hypergraph (see uspgrushgr 40405), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Definitiondf-usgr 40381* Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 40408), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Theoremisuspgr 40382* The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremisusgr 40383* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))

Theoremuspgrf 40384* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremusgrf 40385* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})

Theoremisusgrs 40386* The property of being a simple graph, simplified version of isusgr 40383. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremusgrfs 40387* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 40385. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremusgrfun 40388 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
(𝐺 ∈ USGraph → Fun (iEdg‘𝐺))

Theoremusgrusgra 40389 A simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
(𝐺 ∈ USGraph → (Vtx‘𝐺) USGrph (iEdg‘𝐺))

Theoremusgredgss 40390* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
(𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})

Theoremedgusgr 40391 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐸) = 2))

Theoremisusgrop 40392* The property of being an undirected simple graph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 30-Nov-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremusgrop 40393 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph )

Theoremisausgr 40394* The property of an unordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremausgrusgrb 40395* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ))

Theoremusgrausgri 40396* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       (𝐻 ∈ USGraph → (Vtx‘𝐻)𝐺(Edg‘𝐻))

Theoremausgrumgri 40397* If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph )

Theoremausgrusgri 40398* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph )

Theoremusgrausgrb 40399* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}    &   𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}       ((𝐻𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph ))

Theoremusgredgop 40400 An edge of a simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 15-Oct-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 = (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} ↔ ⟨𝑋, {𝑀, 𝑁}⟩ ∈ 𝐸))

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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 42301-42360
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