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Theorem List for Metamath Proof Explorer - 47401-47500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcspr 47401 Extend class notation with set of pairs.
class Pairs
 
Definitiondf-spr 47402* Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})
 
Theoremsprval 47403* The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprvalpw 47404* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprssspr 47405* The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
(Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
 
Theoremspr0el 47406 The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
∅ ∉ (Pairs‘𝑉)
 
Theoremsprvalpwn0 47407* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprel 47408* An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
 
Theoremprssspr 47409* An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
 
Theoremprelspr 47410 An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
 
Theoremprsprel 47411 The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
 
Theoremprsssprel 47412 The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
 
Theoremsprvalpwle2 47413* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
 
Theoremsprsymrelfvlem 47414* Lemma for sprsymrelf 47419 and sprsymrelfv 47418. (Contributed by AV, 19-Nov-2021.)
(𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
 
Theoremsprsymrelf1lem 47415* Lemma for sprsymrelf1 47420. (Contributed by AV, 22-Nov-2021.)
((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))
 
Theoremsprsymrelfolem1 47416* Lemma 1 for sprsymrelfo 47421. (Contributed by AV, 22-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       𝑄 ∈ 𝒫 (Pairs‘𝑉)
 
Theoremsprsymrelfolem2 47417* Lemma 2 for sprsymrelfo 47421. (Contributed by AV, 23-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
 
Theoremsprsymrelfv 47418* The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
 
Theoremsprsymrelf 47419* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃𝑅
 
Theoremsprsymrelf1 47420* The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃1-1𝑅
 
Theoremsprsymrelfo 47421* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃onto𝑅)
 
Theoremsprsymrelf1o 47422* The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃1-1-onto𝑅)
 
Theoremsprbisymrel 47423* There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)
 
Theoremsprsymrelen 47424* The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝑃𝑅)
 
21.48.11.3  Proper (unordered) pairs

Proper (unordered) pairs are unordered pairs with exactly 2 elements. The set of proper pairs with elements of a class 𝑉 is defined by {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}.

For example, {1, 2} is a proper pair, because 1 ≠ 2 ( see 1ne2 12471). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4866), {1, V} = {1} (see prprc2 4770) or {V, V} = ∅ (see prprc 4771).

 
Theoremprpair 47425* Characterization of a proper pair: A class is a proper pair iff it consists of exactly two different sets. (Contributed by AV, 11-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝑋𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎𝑏))
 
Theoremprproropf1olem0 47426 Lemma 0 for prproropf1o 47431. Remark: 𝑂, the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, can alternatively be written as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ (1st𝑥)𝑅(2nd𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))       (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
 
Theoremprproropf1olem1 47427* Lemma 1 for prproropf1o 47431. (Contributed by AV, 12-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)
 
Theoremprproropf1olem2 47428* Lemma 2 for prproropf1o 47431. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑋𝑃) → ⟨inf(𝑋, 𝑉, 𝑅), sup(𝑋, 𝑉, 𝑅)⟩ ∈ 𝑂)
 
Theoremprproropf1olem3 47429* Lemma 3 for prproropf1o 47431. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)
 
Theoremprproropf1olem4 47430* Lemma 4 for prproropf1o 47431. (Contributed by AV, 14-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))
 
Theoremprproropf1o 47431* There is a bijection between the set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)
 
Theoremprproropen 47432* The set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, are equinumerous. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑉𝑊𝑅 Or 𝑉) → 𝑂𝑃)
 
Theoremprproropreud 47433* There is exactly one ordered ordered pair fulfilling a wff iff there is exactly one proper pair fulfilling an equivalent wff. (Contributed by AV, 20-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   (𝜑𝑅 Or 𝑉)    &   (𝑥 = ⟨inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)⟩ → (𝜓𝜒))    &   (𝑥 = 𝑧 → (𝜓𝜃))       (𝜑 → (∃!𝑥𝑂 𝜓 ↔ ∃!𝑦𝑃 𝜒))
 
Theorempairreueq 47434* Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))
 
Theorempaireqne 47435* Two sets are not equal iff there is exactly one proper pair whose elements are either one of these sets. (Contributed by AV, 27-Jan-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))
 
21.48.11.4  Set of proper unordered pairs
 
Syntaxcprpr 47436 Extend class notation with set of proper unordered pairs.
class Pairsproper
 
Definitiondf-prpr 47437* Define the function which maps a set 𝑣 to the set of proper unordered pairs consisting of exactly two (different) elements of the set 𝑣. (Contributed by AV, 29-Apr-2023.)
Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
 
Theoremprprval 47438* The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
 
Theoremprprvalpw 47439* The set of all proper unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
 
Theoremprprelb 47440 An element of the set of all proper unordered pairs over a given set 𝑉 is a subset of 𝑉 of size two. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (𝑃 ∈ (Pairsproper𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2)))
 
Theoremprprelprb 47441* A set is an element of the set of all proper unordered pairs over a given set 𝑋 iff it is a pair of different elements of the set 𝑋. (Contributed by AV, 7-May-2023.)
(𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))
 
Theoremprprspr2 47442* The set of all proper unordered pairs over a given set 𝑉 is the set of all unordered pairs over that set of size two. (Contributed by AV, 29-Apr-2023.)
(Pairsproper𝑉) = {𝑝 ∈ (Pairs‘𝑉) ∣ (♯‘𝑝) = 2}
 
Theoremprprsprreu 47443* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique unordered pair over 𝑉 of size two fulfilling this wff. (Contributed by AV, 30-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ (Pairs‘𝑉)((♯‘𝑝) = 2 ∧ 𝜑)))
 
Theoremprprreueq 47444* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique subset of 𝑉 of size two fulfilling this wff. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)))
 
Theoremsbcpr 47445* The proper substitution of an unordered pair for a setvar variable corresponds to a proper substitution of each of its elements. (Contributed by AV, 7-Apr-2023.)
(𝑝 = {𝑥, 𝑦} → (𝜑𝜓))       ([{𝑎, 𝑏} / 𝑝]𝜑[𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
 
Theoremreupr 47446* There is a unique unordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 7-Apr-2023.)
(𝑝 = {𝑎, 𝑏} → (𝜓𝜒))    &   (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))       (𝑋𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
 
Theoremreuprpr 47447* There is a unique proper unordered pair fulfilling a wff iff there are uniquely two different sets fulfilling a corresponding wff. (Contributed by AV, 30-Apr-2023.)
(𝑝 = {𝑎, 𝑏} → (𝜓𝜒))    &   (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))       (𝑋𝑉 → (∃!𝑝 ∈ (Pairsproper𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝑎𝑏𝜒 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥𝑦𝜃) → {𝑥, 𝑦} = {𝑎, 𝑏}))))
 
Theorempoprelb 47448 Equality for unordered pairs with partially ordered elements. (Contributed by AV, 9-Jul-2023.)
(((Rel 𝑅𝑅 Po 𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝑅𝐵𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theorem2exopprim 47449 The existence of an ordered pair fulfilling a wff implies the existence of an unordered pair fulfilling the wff. (Contributed by AV, 29-Jul-2023.)
(∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑))
 
Theoremreuopreuprim 47450* There is a unique unordered pair with ordered elements fulfilling a wff if there is a unique ordered pair fulfilling the wff. (Contributed by AV, 28-Jul-2023.)
(𝑋𝑉 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑)))
 
21.48.12  Number theory (extension)
 
21.48.12.1  Fermat numbers

At first, the (sequence of) Fermat numbers FermatNo (the 𝑛-th Fermat number is denoted as (FermatNo‘𝑛)) is defined, see df-fmtno 47452, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 47501, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 47507. The fourth Fermat number (i.e., the fifth Fermat prime) (FermatNo‘4) = 65537 is currently the biggest number proven to be prime in set.mm, see 65537prm 47500 (previously, it was 4001, see 4001prm 17178).

Another important result of this section is Goldbach's theorem goldbachth 47471, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 47512.

Finally, it is shown that every prime of the form ((2↑𝑘) + 1) must be a Fermat number (i.e., a Fermat prime), see 2pwp1prmfmtno 47514.

 
Syntaxcfmtno 47451 Extend class notation with the Fermat numbers.
class FermatNo
 
Definitiondf-fmtno 47452 Define the function that enumerates the Fermat numbers, see definition in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
 
Theoremfmtno 47453 The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))
 
Theoremfmtnoge3 47454 Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ‘3))
 
Theoremfmtnonn 47455 Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ)
 
Theoremfmtnom1nn 47456 A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁)))
 
Theoremfmtnoodd 47457 Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
(𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))
 
Theoremfmtnorn 47458* A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
(𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
 
Theoremfmtnof1 47459 The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.)
FermatNo:ℕ01-1→ℕ
 
Theoremfmtnoinf 47460 The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
ran FermatNo ∉ Fin
 
Theoremfmtnorec1 47461 The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1))
 
Theoremsqrtpwpw2p 47462 The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1))))
 
Theoremfmtnosqrt 47463 The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1))))
 
Theoremfmtno0 47464 The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) = 3
 
Theoremfmtno1 47465 The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) = 5
 
Theoremfmtnorec2lem 47466* Lemma for fmtnorec2 47467 (induction step). (Contributed by AV, 29-Jul-2021.)
(𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))
 
Theoremfmtnorec2 47467* The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))
 
Theoremfmtnodvds 47468 Any Fermat number divides a greater Fermat number minus 2. Corollary of fmtnorec2 47467, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2))
 
Theoremgoldbachthlem1 47469 Lemma 1 for goldbachth 47471. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))
 
Theoremgoldbachthlem2 47470 Lemma 2 for goldbachth 47471. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)
 
Theoremgoldbachth 47471 Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑁𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)
 
Theoremfmtnorec3 47472* The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛))))
 
Theoremfmtnorec4 47473 The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2))))
 
Theoremfmtno2 47474 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17
 
Theoremfmtno3 47475 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257
 
Theoremfmtno4 47476 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537
 
Theoremfmtno5lem1 47477 Lemma 1 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216
 
Theoremfmtno5lem2 47478 Lemma 2 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.)
(65536 · 5) = 327680
 
Theoremfmtno5lem3 47479 Lemma 3 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.)
(65536 · 3) = 196608
 
Theoremfmtno5lem4 47480 Lemma 4 for fmtno5 47481. (Contributed by AV, 30-Jul-2021.)
(65536↑2) = 4294967296
 
Theoremfmtno5 47481 The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
(FermatNo‘5) = 4294967297
 
Theoremfmtno0prm 47482 The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) ∈ ℙ
 
Theoremfmtno1prm 47483 The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) ∈ ℙ
 
Theoremfmtno2prm 47484 The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) ∈ ℙ
 
Theorem257prm 47485 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
257 ∈ ℙ
 
Theoremfmtno3prm 47486 The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
(FermatNo‘3) ∈ ℙ
 
Theoremodz2prm2pw 47487 Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.)
(((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))
 
Theoremfmtnoprmfac1lem 47488 Lemma for fmtnoprmfac1 47489: The order of 2 modulo a prime that divides the n-th Fermat number is 2^(n+1). (Contributed by AV, 25-Jul-2021.) (Proof shortened by AV, 18-Mar-2022.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))
 
Theoremfmtnoprmfac1 47489* Divisor of Fermat number (special form of Euler's result, see fmtnofac1 47494): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
 
Theoremfmtnoprmfac2lem1 47490 Lemma for fmtnoprmfac2 47491. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)
 
Theoremfmtnoprmfac2 47491* Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 47493): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+2)+1 where k is a positive integer. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 2))) + 1))
 
Theoremfmtnofac2lem 47492* Lemma for fmtnofac2 47493 (Induction step). (Contributed by AV, 30-Jul-2021.)
((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((((𝑁 ∈ (ℤ‘2) ∧ 𝑦 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑦 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) ∧ ((𝑁 ∈ (ℤ‘2) ∧ 𝑧 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑧 = ((𝑘 · (2↑(𝑁 + 2))) + 1))) → ((𝑁 ∈ (ℤ‘2) ∧ (𝑦 · 𝑧) ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 (𝑦 · 𝑧) = ((𝑘 · (2↑(𝑁 + 2))) + 1))))
 
Theoremfmtnofac2 47493* Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 47494: Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+2)+1 where k is a nonnegative integer. (Contributed by AV, 30-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 2))) + 1))
 
Theoremfmtnofac1 47494* Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of Fermat Number/Euler's Result", 24-Jul-2021, https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result): "Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+1)+1 where k is a positive integer." Here, however, k must be a nonnegative integer, because k must be 0 to represent 1 (which is a divisor of Fn ).

Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number Fn is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 47493. (Contributed by AV, 30-Jul-2021.)

((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
 
Theoremfmtno4sqrt 47495 The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
(⌊‘(√‘(FermatNo‘4))) = 256
 
Theoremfmtno4prmfac 47496 If P was a (prime) factor of the fourth Fermat number less than the square root of the fourth Fermat number, it would be either 65 or 129 or 193. (Contributed by AV, 28-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = 65 ∨ 𝑃 = 129 ∨ 𝑃 = 193))
 
Theoremfmtno4prmfac193 47497 If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = 193)
 
Theoremfmtno4nprmfac193 47498 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
¬ 193 ∥ (FermatNo‘4)
 
Theoremfmtno4prm 47499 The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
(FermatNo‘4) ∈ ℙ
 
Theorem65537prm 47500 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
65537 ∈ ℙ
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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 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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49035
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