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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cspr 47401 | Extend class notation with set of pairs. |
class Pairs | ||
Definition | df-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 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprval 47403* | The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprvalpw 47404* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprssspr 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 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
Theorem | spr0el 47406 | The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ ∅ ∉ (Pairs‘𝑉) | ||
Theorem | sprvalpwn0 47407* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprel 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) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
Theorem | prssspr 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) ∧ 𝑋 ∈ 𝑃) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
Theorem | prelspr 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.) |
⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) | ||
Theorem | prsprel 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) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
Theorem | prsssprel 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 ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
Theorem | sprvalpwle2 47413* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) | ||
Theorem | sprsymrelfvlem 47414* | Lemma for sprsymrelf 47419 and sprsymrelfv 47418. (Contributed by AV, 19-Nov-2021.) |
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) | ||
Theorem | sprsymrelf1lem 47415* | Lemma for sprsymrelf1 47420. (Contributed by AV, 22-Nov-2021.) |
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏)) | ||
Theorem | sprsymrelfolem1 47416* | Lemma 1 for sprsymrelfo 47421. (Contributed by AV, 22-Nov-2021.) |
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) | ||
Theorem | sprsymrelfolem2 47417* | Lemma 2 for sprsymrelfo 47421. (Contributed by AV, 23-Nov-2021.) |
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})) | ||
Theorem | sprsymrelfv 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) | ||
Theorem | sprsymrelf 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃⟶𝑅 | ||
Theorem | sprsymrelf1 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃–1-1→𝑅 | ||
Theorem | sprsymrelfo 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) | ||
Theorem | sprsymrelf1o 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) | ||
Theorem | sprbisymrel 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅) | ||
Theorem | sprsymrelen 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) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅) | ||
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). | ||
Theorem | prpair 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} ⇒ ⊢ (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏)) | ||
Theorem | prproropf1olem0 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) ∣ (1st ‘𝑥)𝑅(2nd ‘𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ 〈(1st ‘𝑥), (2nd ‘𝑥)〉 ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.) |
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) ⇒ ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) | ||
Theorem | prproropf1olem1 47427* | Lemma 1 for prproropf1o 47431. (Contributed by AV, 12-Mar-2023.) |
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) & ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} ⇒ ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → {(1st ‘𝑊), (2nd ‘𝑊)} ∈ 𝑃) | ||
Theorem | prproropf1olem2 47428* | Lemma 2 for prproropf1o 47431. (Contributed by AV, 13-Mar-2023.) |
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) & ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} ⇒ ⊢ ((𝑅 Or 𝑉 ∧ 𝑋 ∈ 𝑃) → 〈inf(𝑋, 𝑉, 𝑅), sup(𝑋, 𝑉, 𝑅)〉 ∈ 𝑂) | ||
Theorem | prproropf1olem3 47429* | Lemma 3 for prproropf1o 47431. (Contributed by AV, 13-Mar-2023.) |
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) & ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) ⇒ ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑂) → (𝐹‘{(1st ‘𝑊), (2nd ‘𝑊)}) = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | ||
Theorem | prproropf1olem4 47430* | Lemma 4 for prproropf1o 47431. (Contributed by AV, 14-Mar-2023.) |
⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) & ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) ⇒ ⊢ ((𝑅 Or 𝑉 ∧ 𝑊 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → ((𝐹‘𝑍) = (𝐹‘𝑊) → 𝑍 = 𝑊)) | ||
Theorem | prproropf1o 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 = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) ⇒ ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂) | ||
Theorem | prproropen 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} ⇒ ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 Or 𝑉) → 𝑂 ≈ 𝑃) | ||
Theorem | prproropreud 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} & ⊢ (𝜑 → 𝑅 Or 𝑉) & ⊢ (𝑥 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) | ||
Theorem | pairreueq 47434* | Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.) |
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (∃!𝑝 ∈ 𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)) | ||
Theorem | paireqne 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⇒ ⊢ (𝜑 → (∃!𝑝 ∈ 𝑃 ∀𝑥 ∈ 𝑝 (𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ↔ 𝐴 ≠ 𝐵)) | ||
Syntax | cprpr 47436 | Extend class notation with set of proper unordered pairs. |
class Pairsproper | ||
Definition | df-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 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}) | ||
Theorem | prprval 47438* | The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.) |
⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}) | ||
Theorem | prprvalpw 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.) |
⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})}) | ||
Theorem | prprelb 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.) |
⊢ (𝑉 ∈ 𝑊 → (𝑃 ∈ (Pairsproper‘𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2))) | ||
Theorem | prprelprb 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 ∈ (Pairsproper‘𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))) | ||
Theorem | prprspr2 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.) |
⊢ (Pairsproper‘𝑉) = {𝑝 ∈ (Pairs‘𝑉) ∣ (♯‘𝑝) = 2} | ||
Theorem | prprsprreu 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.) |
⊢ (𝑉 ∈ 𝑊 → (∃!𝑝 ∈ (Pairsproper‘𝑉)𝜑 ↔ ∃!𝑝 ∈ (Pairs‘𝑉)((♯‘𝑝) = 2 ∧ 𝜑))) | ||
Theorem | prprreueq 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.) |
⊢ (𝑉 ∈ 𝑊 → (∃!𝑝 ∈ (Pairsproper‘𝑉)𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))) | ||
Theorem | sbcpr 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} → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓) | ||
Theorem | reupr 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} → (𝜓 ↔ 𝜒)) & ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))) | ||
Theorem | reuprpr 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} → (𝜓 ↔ 𝜒)) & ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (Pairsproper‘𝑋)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 ∧ 𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 ≠ 𝑦 ∧ 𝜃) → {𝑥, 𝑦} = {𝑎, 𝑏})))) | ||
Theorem | poprelb 47448 | Equality for unordered pairs with partially ordered elements. (Contributed by AV, 9-Jul-2023.) |
⊢ (((Rel 𝑅 ∧ 𝑅 Po 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝑅𝐵 ∧ 𝐶𝑅𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | 2exopprim 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.) |
⊢ (∃𝑎∃𝑏(〈𝐴, 𝐵〉 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)) | ||
Theorem | reuopreuprim 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.) |
⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ 𝜑) → ∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑))) | ||
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. | ||
Syntax | cfmtno 47451 | Extend class notation with the Fermat numbers. |
class FermatNo | ||
Definition | df-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)) | ||
Theorem | fmtno 47453 | The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | ||
Theorem | fmtnoge3 47454 | Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) | ||
Theorem | fmtnonn 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) | ||
Theorem | fmtnom1nn 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))) | ||
Theorem | fmtnoodd 47457 | Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.) |
⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) | ||
Theorem | fmtnorn 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) | ||
Theorem | fmtnof1 47459 | The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.) |
⊢ FermatNo:ℕ0–1-1→ℕ | ||
Theorem | fmtnoinf 47460 | The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.) |
⊢ ran FermatNo ∉ Fin | ||
Theorem | fmtnorec1 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)) | ||
Theorem | sqrtpwpw2p 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 ∈ ℕ0 ∧ 𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1)))) | ||
Theorem | fmtnosqrt 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)))) | ||
Theorem | fmtno0 47464 | The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘0) = 3 | ||
Theorem | fmtno1 47465 | The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘1) = 5 | ||
Theorem | fmtnorec2lem 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))) | ||
Theorem | fmtnorec2 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)) | ||
Theorem | fmtnodvds 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 ∈ ℕ0 ∧ 𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2)) | ||
Theorem | goldbachthlem1 47469 | Lemma 1 for goldbachth 47471. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2)) | ||
Theorem | goldbachthlem2 47470 | Lemma 2 for goldbachth 47471. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) | ||
Theorem | goldbachth 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 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) | ||
Theorem | fmtnorec3 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)))) | ||
Theorem | fmtnorec4 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)))) | ||
Theorem | fmtno2 47474 | The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘2) = ;17 | ||
Theorem | fmtno3 47475 | The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘3) = ;;257 | ||
Theorem | fmtno4 47476 | The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘4) = ;;;;65537 | ||
Theorem | fmtno5lem1 47477 | Lemma 1 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 6) = ;;;;;393216 | ||
Theorem | fmtno5lem2 47478 | Lemma 2 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 5) = ;;;;;327680 | ||
Theorem | fmtno5lem3 47479 | Lemma 3 for fmtno5 47481. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 3) = ;;;;;196608 | ||
Theorem | fmtno5lem4 47480 | Lemma 4 for fmtno5 47481. (Contributed by AV, 30-Jul-2021.) |
⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296 | ||
Theorem | fmtno5 47481 | The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.) |
⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 | ||
Theorem | fmtno0prm 47482 | The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘0) ∈ ℙ | ||
Theorem | fmtno1prm 47483 | The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘1) ∈ ℙ | ||
Theorem | fmtno2prm 47484 | The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.) |
⊢ (FermatNo‘2) ∈ ℙ | ||
Theorem | 257prm 47485 | 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.) |
⊢ ;;257 ∈ ℙ | ||
Theorem | fmtno3prm 47486 | The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.) |
⊢ (FermatNo‘3) ∈ ℙ | ||
Theorem | odz2prm2pw 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)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) | ||
Theorem | fmtnoprmfac1lem 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)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) | ||
Theorem | fmtnoprmfac1 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)) | ||
Theorem | fmtnoprmfac2lem1 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) | ||
Theorem | fmtnoprmfac2 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)) | ||
Theorem | fmtnofac2lem 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)))) | ||
Theorem | fmtnofac2 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)) | ||
Theorem | fmtnofac1 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)) | ||
Theorem | fmtno4sqrt 47495 | The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.) |
⊢ (⌊‘(√‘(FermatNo‘4))) = ;;256 | ||
Theorem | fmtno4prmfac 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)) | ||
Theorem | fmtno4prmfac193 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) | ||
Theorem | fmtno4nprmfac193 47498 | 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.) |
⊢ ¬ ;;193 ∥ (FermatNo‘4) | ||
Theorem | fmtno4prm 47499 | The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.) |
⊢ (FermatNo‘4) ∈ ℙ | ||
Theorem | 65537prm 47500 | 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.) |
⊢ ;;;;65537 ∈ ℙ |
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