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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhashunx 12901 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 12897. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (#‘(𝐴𝐵)) = ((#‘𝐴) +𝑒 (#‘𝐵)))
 
Theoremhashge0 12902 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → 0 ≤ (#‘𝐴))
 
Theoremhashgt0 12903 The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 0 < (#‘𝐴))
 
Theoremhashge1 12904 The cardinality of a nonempty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 1 ≤ (#‘𝐴))
 
Theorem1elfz0hash 12905 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(#‘𝐴)))
 
Theoremhashnn0n0nn 12906 If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.)
(((𝑉𝑊𝑌 ∈ ℕ0) ∧ ((#‘𝑉) = 𝑌𝑁𝑉)) → 𝑌 ∈ ℕ)
 
Theoremhashunsng 12907 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
(𝐵𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵𝐴) → (#‘(𝐴 ∪ {𝐵})) = ((#‘𝐴) + 1)))
 
Theoremhashprg 12908 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
 
TheoremhashprgOLD 12909 Obsolete version of hashprg 12908 as of 18-Sep-2021. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) TODO-AV: to be removed after revision of graph theory! (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑉) → (𝐴𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
 
Theoremelprchashprn2 12910 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
𝑀 ∈ V → ¬ (#‘{𝑀, 𝑁}) = 2)
 
Theoremhashprb 12911 The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀𝑁) ↔ (#‘{𝑀, 𝑁}) = 2)
 
Theoremhashprdifel 12912 The elements of an unordered pair of size 2 are different sets. (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}       ((#‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
 
Theoremprhash2ex 12913 There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 12922, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
(#‘{0, 1}) = 2
 
Theoremhashle00 12914 If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
(𝑉𝑊 → ((#‘𝑉) ≤ 0 ↔ 𝑉 = ∅))
 
Theoremhashgt0elex 12915* If the size of a set is greater than zero, the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑉𝑊 ∧ 0 < (#‘𝑉)) → ∃𝑥 𝑥𝑉)
 
Theoremhashgt0elexb 12916* The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018.)
(𝑉𝑊 → (0 < (#‘𝑉) ↔ ∃𝑥 𝑥𝑉))
 
Theoremhashp1i 12917 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝐴 ∈ ω    &   𝐵 = suc 𝐴    &   (#‘𝐴) = 𝑀    &   (𝑀 + 1) = 𝑁       (#‘𝐵) = 𝑁
 
Theoremhash1 12918 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘1𝑜) = 1
 
Theoremhash2 12919 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘2𝑜) = 2
 
Theoremhash3 12920 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘3𝑜) = 3
 
Theoremhash4 12921 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘4𝑜) = 4
 
Theorempr0hash2ex 12922 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
(#‘{∅, {∅}}) = 2
 
Theoremhashss 12923 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝐴𝑉𝐵𝐴) → (#‘𝐵) ≤ (#‘𝐴))
 
Theoremprsshashgt1 12924 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
(((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐶𝑈) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (#‘𝐶)))
 
Theoremhashin 12925 The size of the intersection of a set and a class is less than or equal to the size of the set. (Contributed by AV, 4-Jan-2021.)
(𝐴𝑉 → (#‘(𝐴𝐵)) ≤ (#‘𝐴))
 
Theoremhashssdif 12926 The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘(𝐴𝐵)) = ((#‘𝐴) − (#‘𝐵)))
 
Theoremhashdif 12927 The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.)
(𝐴 ∈ Fin → (#‘(𝐴𝐵)) = ((#‘𝐴) − (#‘(𝐴𝐵))))
 
Theoremhashdifsn 12928 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘(𝐴 ∖ {𝐵})) = ((#‘𝐴) − 1))
 
Theoremhashdifpr 12929 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
((𝐴 ∈ Fin ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (#‘(𝐴 ∖ {𝐵, 𝐶})) = ((#‘𝐴) − 2))
 
Theoremhashsn01 12930 The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
((#‘{𝐴}) = 0 ∨ (#‘{𝐴}) = 1)
 
Theoremhashsnle1 12931 The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
(#‘{𝐴}) ≤ 1
 
Theoremhashsnlei 12932 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.) (Proof shortened by AV, 23-Feb-2021.)
({𝐴} ∈ Fin ∧ (#‘{𝐴}) ≤ 1)
 
Theoremhash1snb 12933* The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.)
(𝑉𝑊 → ((#‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))
 
Theoremeuhash1 12934* The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
(𝑉𝑊 → ((#‘𝑉) = 1 ↔ ∃!𝑎 𝑎𝑉))
 
Theoremhashgt12el 12935* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (#‘𝑉)) → ∃𝑎𝑉𝑏𝑉 𝑎𝑏)
 
Theoremhashgt12el2 12936* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (#‘𝑉) ∧ 𝐴𝑉) → ∃𝑏𝑉 𝐴𝑏)
 
Theoremhashunlei 12937 Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐶 = (𝐴𝐵)    &   (𝐴 ∈ Fin ∧ (#‘𝐴) ≤ 𝐾)    &   (𝐵 ∈ Fin ∧ (#‘𝐵) ≤ 𝑀)    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐾 + 𝑀) = 𝑁       (𝐶 ∈ Fin ∧ (#‘𝐶) ≤ 𝑁)
 
Theoremhashsslei 12938 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐵𝐴    &   (𝐴 ∈ Fin ∧ (#‘𝐴) ≤ 𝑁)    &   𝑁 ∈ ℕ0       (𝐵 ∈ Fin ∧ (#‘𝐵) ≤ 𝑁)
 
Theoremhashfz 12939 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
(𝐵 ∈ (ℤ𝐴) → (#‘(𝐴...𝐵)) = ((𝐵𝐴) + 1))
 
Theoremfzsdom2 12940 Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)
(((𝐵 ∈ (ℤ𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶))
 
Theoremhashfzo 12941 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (#‘(𝐴..^𝐵)) = (𝐵𝐴))
 
Theoremhashfzo0 12942 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ ℕ0 → (#‘(0..^𝐵)) = 𝐵)
 
Theoremhashfzp1 12943 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))
 
Theoremhashfz0 12944 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝐵 ∈ ℕ0 → (#‘(0...𝐵)) = (𝐵 + 1))
 
Theoremhashxplem 12945 Lemma for hashxp 12946. (Contributed by Paul Chapman, 30-Nov-2012.)
𝐵 ∈ Fin       (𝐴 ∈ Fin → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))
 
Theoremhashxp 12946 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))
 
Theoremhashmap 12947 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝑚 𝐵)) = ((#‘𝐴)↑(#‘𝐵)))
 
Theoremhashpw 12948 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)
(𝐴 ∈ Fin → (#‘𝒫 𝐴) = (2↑(#‘𝐴)))
 
Theoremhashfun 12949 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)
(𝐹 ∈ Fin → (Fun 𝐹 ↔ (#‘𝐹) = (#‘dom 𝐹)))
 
Theoremhashimarn 12950 The size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 equals the size of the function 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹)))
 
Theoremhashimarni 12951 If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → (#‘𝐹) = 𝑁))
 
Theoremfnfz0hash 12952 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0...𝑁)) → (#‘𝐹) = (𝑁 + 1))
 
Theoremffz0hash 12953 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶𝐵) → (#‘𝐹) = (𝑁 + 1))
 
Theoremfnfz0hashnn0 12954 The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0...𝑁) → (#‘𝐹) ∈ ℕ0)
 
Theoremffzo0hash 12955 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0..^𝑁)) → (#‘𝐹) = 𝑁)
 
Theoremfnfzo0hash 12956 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0..^𝑁)⟶𝐵) → (#‘𝐹) = 𝑁)
 
Theoremfnfzo0hashnn0 12957 The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0..^𝑁) → (#‘𝐹) ∈ ℕ0)
 
Theoremhashbclem 12958* Lemma for hashbc 12959: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ∀𝑗 ∈ ℤ ((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}))    &   (𝜑𝐾 ∈ ℤ)       (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾}))
 
Theoremhashbc 12959* The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)
((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
 
Theoremhashfacen 12960* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴𝐵𝐶𝐷) → {𝑓𝑓:𝐴1-1-onto𝐶} ≈ {𝑓𝑓:𝐵1-1-onto𝐷})
 
Theoremhashf1lem1 12961* Lemma for hashf1 12963. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))    &   (𝜑𝐹:𝐴1-1𝐵)       (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
 
Theoremhashf1lem2 12962* Lemma for hashf1 12963. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))       (𝜑 → (#‘{𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓𝑓:𝐴1-1𝐵})))
 
Theoremhashf1 12963* The permutation number 𝐴 ∣ ! · ( ∣ 𝐵 ∣ C ∣ 𝐴 ∣ ) = 𝐵 ∣ ! / ( ∣ 𝐵 ∣ − ∣ 𝐴 ∣ )! counts the number of injections from 𝐴 to 𝐵. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘{𝑓𝑓:𝐴1-1𝐵}) = ((!‘(#‘𝐴)) · ((#‘𝐵)C(#‘𝐴))))
 
Theoremhashfac 12964* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)
(𝐴 ∈ Fin → (#‘{𝑓𝑓:𝐴1-1-onto𝐴}) = (!‘(#‘𝐴)))
 
Theoremleiso 12965 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
 
Theoremleisorel 12966 Version of isorel 6353 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
 
Theoremfz1isolem 12967* Lemma for fz1iso 12968. (Contributed by Mario Carneiro, 2-Apr-2014.)
𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)    &   𝐵 = (ℕ ∩ ( < “ {((#‘𝐴) + 1)}))    &   𝐶 = (ω ∩ (𝐺‘((#‘𝐴) + 1)))    &   𝑂 = OrdIso(𝑅, 𝐴)       ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))
 
Theoremfz1iso 12968* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))
 
Theoremishashinf 12969* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7934. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)
 
Theoremseqcoll 12970* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝑁 ∈ (1...(#‘𝐴)))    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))
 
Theoremseqcoll2 12971* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
 
5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)
 
Theoremhashprlei 12972 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)
 
Theoremhash2pr 12973* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏 𝑉 = {𝑎, 𝑏})
 
Theoremhash2prde 12974* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏}))
 
Theoremhash2exprb 12975* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏})))
 
Theoremhash2prb 12976* A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
 
Theoremhash2prd 12977 A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 1-Nov-2020.)
((𝑃𝑉 ∧ (#‘𝑃) = 2) → ((𝑋𝑃𝑌𝑃𝑋𝑌) → 𝑃 = {𝑋, 𝑌}))
 
Theoremhash2pwpr 12978 If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)
(((#‘𝑃) = 2 ∧ 𝑃 ∈ 𝒫 {𝑋, 𝑌}) → 𝑃 = {𝑋, 𝑌})
 
Theorempr2pwpr 12979* The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2𝑜} = {{𝐴, 𝐵}})
 
Theoremhashge2el2dif 12980* A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)
((𝐷𝑉 ∧ 2 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷 𝑥𝑦)
 
Theoremhashge2el2difr 12981* A set with at least 2 different elements has size at least 2. (Contributed by AV, 14-Oct-2020.)
((𝐷𝑉 ∧ ∃𝑥𝐷𝑦𝐷 𝑥𝑦) → 2 ≤ (#‘𝐷))
 
Theoremhashge2el2difb 12982* A set has size at least 2 iff it has at least 2 different elements. (Contributed by AV, 14-Oct-2020.)
(𝐷𝑉 → (2 ≤ (#‘𝐷) ↔ ∃𝑥𝐷𝑦𝐷 𝑥𝑦))
 
Theoremhashtplei 12983 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵, 𝐶} ∈ Fin ∧ (#‘{𝐴, 𝐵, 𝐶}) ≤ 3)
 
Theoremhashtpg 12984 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑈𝐵𝑉𝐶𝑊) → ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (#‘{𝐴, 𝐵, 𝐶}) = 3))
 
Theoremhashge3el3dif 12985* A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 12980, which has an elementary proof, the dominance relation and 1-1 functions from a set with three elements which are known to be different are used to prove this theorem. Although there is also an elementary proof for this theorem, it might be much longer. After all, this proof should be kept because it can be used as template for proofs for higher cardinalities. (Contributed by AV, 20-Mar-2019.) (Proof modification is discouraged.)
((𝐷𝑉 ∧ 3 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷𝑧𝐷 (𝑥𝑦𝑥𝑧𝑦𝑧))
 
Theoremelss2prb 12986* An element of the set of subsets with two elements is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑃 ∈ {𝑧 ∈ 𝒫 𝑉 ∣ (#‘𝑧) = 2} ↔ ∃𝑥𝑉𝑦𝑉 (𝑥𝑦𝑃 = {𝑥, 𝑦}))
 
Theoremhash2sspr 12987* A subset of size two is an unordered pair of elements of its superset. (Contributed by Alexander van der Vekens, 12-Jul-2017.) (Proof shortened by AV, 4-Nov-2020.)
((𝑃 ∈ 𝒫 𝑉 ∧ (#‘𝑃) = 2) → ∃𝑎𝑉𝑏𝑉 𝑃 = {𝑎, 𝑏})
 
Theoremelss2prOLD 12988* An element of the set of subsets with two elements is an unordered pair. (Contributed by Alexander van der Vekens, 12-Jul-2018.) Obsolete version of elss2prb 12986 as of 1-Nov-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝑃 ∈ {𝑧 ∈ 𝒫 𝑉 ∣ (#‘𝑧) = 2} → ∃𝑥𝑉𝑦𝑉 𝑃 = {𝑥, 𝑦})
 
Theoremexprelprel 12989* If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
(∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝𝑋 → ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ 𝑋)
 
Theoremhash3tr 12990* A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑉𝑊 ∧ (#‘𝑉) = 3) → ∃𝑎𝑏𝑐 𝑉 = {𝑎, 𝑏, 𝑐})
 
Theoremhash1to3 12991* If the size of a set is between 1 and 3 (inclusively), the set is a singleton or an unordered pair or an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3) → ∃𝑎𝑏𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))
 
5.6.11.2  Finite induction on the size of the first component of a binary relation
 
Theorembrfi1indlem 12992 Lemma for brfi1ind 12995: The size of a set is the size of this set with one element removed, increased by 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
((𝑉𝑊𝑁𝑉𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → (#‘(𝑉 ∖ {𝑁})) = 𝑌))
 
Theoremfi1uzind 12993* Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 12998) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 
Theorembrfi1uzind 12994* Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with 𝐿 = 0 (see brfi1ind 12995) or 𝐿 = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.)
Rel 𝐺    &   𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 
Theorembrfi1ind 12995* Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, e.g. usgrafis 25682. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Revised by AV, 28-Mar-2021.)
Rel 𝐺    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
 
Theorembrfi1indALT 12996* Alternate proof of brfi1ind 12995, which does not use brfi1uzind 12994. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Rel 𝐺    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
 
Theoremopfi1uzind 12997* Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 12998) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐸 ∈ V    &   𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 
Theoremopfi1ind 12998* Properties of an ordered pair with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, e.g. fusgrfis 40654. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐸 ∈ V    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → 𝜑)
 
Theoremfi1uzindOLD 12999* Obsolete version of fi1uzind 12993 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 
Theorembrfi1uzindOLD 13000* Obsolete version of brfi1uzind 12994 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
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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 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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 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