P. 454 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45301-45400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sprvalpwle2 45301*	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 45302*	Lemma for sprsymrelf 45307 and sprsymrelfv 45306. (Contributed by AV, 19-Nov-2021.)
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
Theorem	sprsymrelf1lem 45303*	Lemma for sprsymrelf1 45308. (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 45304*	Lemma 1 for sprsymrelfo 45309. (Contributed by AV, 22-Nov-2021.)
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}    ⇒   ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)
Theorem	sprsymrelfolem2 45305*	Lemma 2 for sprsymrelfo 45309. (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 45306*	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 45307*	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 45308*	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 45309*	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 45310*	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 45311*	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 45312*	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)
21.42.11.3  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 12282). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4806), {1, V} = {1} (see prprc2 4714) or {V, V} = ∅ (see prprc 4715).
Theorem	prpair 45313*	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 45314	Lemma 0 for prproropf1o 45319. 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 45315*	Lemma 1 for prproropf1o 45319. (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 45316*	Lemma 2 for prproropf1o 45319. (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 45317*	Lemma 3 for prproropf1o 45319. (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 45318*	Lemma 4 for prproropf1o 45319. (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 45319*	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 45320*	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 45321*	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 45322*	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 45323*	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))
21.42.11.4  Set of proper unordered pairs
Syntax	cprpr 45324	Extend class notation with set of proper unordered pairs.
class Pairsproper
Definition	df-prpr 45325*	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 45326*	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 45327*	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 45328	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 45329*	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 45330*	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 45331*	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 45332*	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 45333*	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 45334*	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 45335*	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 45336	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 45337	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 45338*	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)))
21.42.12  Number theory (extension)
21.42.12.1  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 45340, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 45389, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 45395. 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 45388 (previously, it was ;;;4001, see 4001prm 16943). Another important result of this section is Goldbach's theorem goldbachth 45359, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 45400. 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 45402.
Syntax	cfmtno 45339	Extend class notation with the Fermat numbers.
class FermatNo
Definition	df-fmtno 45340	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 45341	The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))
Theorem	fmtnoge3 45342	Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3))
Theorem	fmtnonn 45343	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 45344	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 45345	Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))
Theorem	fmtnorn 45346*	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 45347	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 45348	The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
⊢ ran FermatNo ∉ Fin
Theorem	fmtnorec1 45349	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 45350	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 45351	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 45352	The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘0) = 3
Theorem	fmtno1 45353	The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘1) = 5
Theorem	fmtnorec2lem 45354*	Lemma for fmtnorec2 45355 (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 45355*	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 45356	Any Fermat number divides a greater Fermat number minus 2. Corollary of fmtnorec2 45355, 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 45357	Lemma 1 for goldbachth 45359. (Contributed by AV, 1-Aug-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))
Theorem	goldbachthlem2 45358	Lemma 2 for goldbachth 45359. (Contributed by AV, 1-Aug-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)
Theorem	goldbachth 45359	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 45360*	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 45361	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 45362	The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) = ;17
Theorem	fmtno3 45363	The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘3) = ;;257
Theorem	fmtno4 45364	The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘4) = ;;;;65537
Theorem	fmtno5lem1 45365	Lemma 1 for fmtno5 45369. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 6) = ;;;;;393216
Theorem	fmtno5lem2 45366	Lemma 2 for fmtno5 45369. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 5) = ;;;;;327680
Theorem	fmtno5lem3 45367	Lemma 3 for fmtno5 45369. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 3) = ;;;;;196608
Theorem	fmtno5lem4 45368	Lemma 4 for fmtno5 45369. (Contributed by AV, 30-Jul-2021.)
⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
Theorem	fmtno5 45369	The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
⊢ (FermatNo‘5) = ;;;;;;;;;4294967297
Theorem	fmtno0prm 45370	The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘0) ∈ ℙ
Theorem	fmtno1prm 45371	The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘1) ∈ ℙ
Theorem	fmtno2prm 45372	The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) ∈ ℙ
Theorem	257prm 45373	257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ ;;257 ∈ ℙ
Theorem	fmtno3prm 45374	The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ (FermatNo‘3) ∈ ℙ
Theorem	odz2prm2pw 45375	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 45376	Lemma for fmtnoprmfac1 45377: 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 45377*	Divisor of Fermat number (special form of Euler's result, see fmtnofac1 45382): 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 45378	Lemma for fmtnoprmfac2 45379. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)
Theorem	fmtnoprmfac2 45379*	Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 45381): 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 45380*	Lemma for fmtnofac2 45381 (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 45381*	Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 45382: 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 45382*	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 45381. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
Theorem	fmtno4sqrt 45383	The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
⊢ (⌊‘(√‘(FermatNo‘4))) = ;;256
Theorem	fmtno4prmfac 45384	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 45385	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 45386	193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
⊢ ¬ ;;193 ∥ (FermatNo‘4)
Theorem	fmtno4prm 45387	The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ (FermatNo‘4) ∈ ℙ
Theorem	65537prm 45388	65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ ;;;;65537 ∈ ℙ
Theorem	fmtnofz04prm 45389	The first five Fermat numbers are prime, see remark in [ApostolNT] p. 7. (Contributed by AV, 28-Jul-2021.)
⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ)
Theorem	fmtnole4prm 45390	The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)
Theorem	fmtno5faclem1 45391	Lemma 1 for fmtno5fac 45394. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668
Theorem	fmtno5faclem2 45392	Lemma 2 for fmtno5fac 45394. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502
Theorem	fmtno5faclem3 45393	Lemma 3 for fmtno5fac 45394. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = ;;;;;;;;428826688
Theorem	fmtno5fac 45394	The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.)
⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641)
Theorem	fmtno5nprm 45395	The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
⊢ (FermatNo‘5) ∉ ℙ
Theorem	prmdvdsfmtnof1lem1 45396*	Lemma 1 for prmdvdsfmtnof1 45399. (Contributed by AV, 3-Aug-2021.)
⊢ 𝐼 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐹}, ℝ, < )    &   ⊢ 𝐽 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐺}, ℝ, < )    ⇒   ⊢ ((𝐹 ∈ (ℤ≥‘2) ∧ 𝐺 ∈ (ℤ≥‘2)) → (𝐼 = 𝐽 → (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)))
Theorem	prmdvdsfmtnof1lem2 45397	Lemma 2 for prmdvdsfmtnof1 45399. (Contributed by AV, 3-Aug-2021.)
⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
Theorem	prmdvdsfmtnof 45398*	The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.) (Proof shortened by II, 16-Feb-2023.)
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ))    ⇒   ⊢ 𝐹:ran FermatNo⟶ℙ
Theorem	prmdvdsfmtnof1 45399*	The mapping of a Fermat number to its smallest prime factor is a one-to-one function. (Contributed by AV, 4-Aug-2021.)
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ))    ⇒   ⊢ 𝐹:ran FermatNo–1-1→ℙ
Theorem	prminf2 45400	The set of prime numbers is infinite. The proof of this variant of prminf 16713 is based on Goldbach's theorem goldbachth 45359 (via prmdvdsfmtnof1 45399 and prmdvdsfmtnof1lem2 45397), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties 45397. (Contributed by AV, 4-Aug-2021.)
⊢ ℙ ∉ Fin