P. 450 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44901-45000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ichexmpl2 44901*	Example for interchangeable setvar variables in an arithmetic expression. (Contributed by AV, 31-Jul-2023.)
⊢ [𝑎⇄𝑏]((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝑎↑2) + (𝑏↑2)) = (𝑐↑2))
Theorem	ich2exprop 44902*	If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
Theorem	ichnreuop 44903*	If the setvar variables are interchangeable in a wff, there is never a unique ordered pair with different components fulfilling the wff (because if ⟨𝑎, 𝑏⟩ fulfils the wff, then also ⟨𝑏, 𝑎⟩ fulfils the wff). (Contributed by AV, 27-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 → ¬ ∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑎 ≠ 𝑏 ∧ 𝜑))
Theorem	ichreuopeq 44904*	If the setvar variables are interchangeable in a wff, and there is a unique ordered pair fulfilling the wff, then both setvar variables must be equal. (Contributed by AV, 28-Aug-2023.)
⊢ ([𝑎⇄𝑏]𝜑 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
20.41.11.2  Set of unordered pairs
Theorem	sprid 44905	Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
Theorem	elsprel 44906*	An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4709, which is not an element of all unordered pairs, see spr0nelg 44907. (Contributed by AV, 21-Nov-2021.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
Theorem	spr0nelg 44907*	The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
Syntax	cspr 44908	Extend class notation with set of pairs.
class Pairs
Definition	df-spr 44909*	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 44910*	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 44911*	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 44912*	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 44913	The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
⊢ ∅ ∉ (Pairs‘𝑉)
Theorem	sprvalpwn0 44914*	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 44915*	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 44916*	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 44917	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 44918	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 44919	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 44920*	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 44921*	Lemma for sprsymrelf 44926 and sprsymrelfv 44925. (Contributed by AV, 19-Nov-2021.)
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
Theorem	sprsymrelf1lem 44922*	Lemma for sprsymrelf1 44927. (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 44923*	Lemma 1 for sprsymrelfo 44928. (Contributed by AV, 22-Nov-2021.)
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}    ⇒   ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)
Theorem	sprsymrelfolem2 44924*	Lemma 2 for sprsymrelfo 44928. (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 44925*	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 44926*	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 44927*	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 44928*	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 44929*	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 44930*	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 44931*	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)
20.41.11.3  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 12192). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4798), {1, V} = {1} (see prprc2 4708) or {V, V} = ∅ (see prprc 4709).
Theorem	prpair 44932*	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 44933	Lemma 0 for prproropf1o 44938. 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 44934*	Lemma 1 for prproropf1o 44938. (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 44935*	Lemma 2 for prproropf1o 44938. (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 44936*	Lemma 3 for prproropf1o 44938. (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 44937*	Lemma 4 for prproropf1o 44938. (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 44938*	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 44939*	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 44940*	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 44941*	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 44942*	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))
20.41.11.4  Set of proper unordered pairs
Syntax	cprpr 44943	Extend class notation with set of proper unordered pairs.
class Pairsproper
Definition	df-prpr 44944*	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 44945*	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 44946*	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 44947	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 44948*	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 44949*	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 44950*	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 44951*	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 44952*	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 44953*	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 44954*	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 44955	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 44956	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 44957*	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)))
20.41.12  Number theory (extension)
20.41.12.1  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 44959, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 45008, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 45014. 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 45007 (previously, it was ;;;4001, see 4001prm 16857). Another important result of this section is Goldbach's theorem goldbachth 44978, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 45019. 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 45021.
Syntax	cfmtno 44958	Extend class notation with the Fermat numbers.
class FermatNo
Definition	df-fmtno 44959	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 44960	The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))
Theorem	fmtnoge3 44961	Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3))
Theorem	fmtnonn 44962	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 44963	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 44964	Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))
Theorem	fmtnorn 44965*	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 44966	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 44967	The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
⊢ ran FermatNo ∉ Fin
Theorem	fmtnorec1 44968	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 44969	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 44970	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 44971	The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘0) = 3
Theorem	fmtno1 44972	The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘1) = 5
Theorem	fmtnorec2lem 44973*	Lemma for fmtnorec2 44974 (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 44974*	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 44975	Any Fermat number divides a greater Fermat number minus 2. Corollary of fmtnorec2 44974, 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 44976	Lemma 1 for goldbachth 44978. (Contributed by AV, 1-Aug-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))
Theorem	goldbachthlem2 44977	Lemma 2 for goldbachth 44978. (Contributed by AV, 1-Aug-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)
Theorem	goldbachth 44978	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 44979*	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 44980	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 44981	The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) = ;17
Theorem	fmtno3 44982	The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘3) = ;;257
Theorem	fmtno4 44983	The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘4) = ;;;;65537
Theorem	fmtno5lem1 44984	Lemma 1 for fmtno5 44988. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 6) = ;;;;;393216
Theorem	fmtno5lem2 44985	Lemma 2 for fmtno5 44988. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 5) = ;;;;;327680
Theorem	fmtno5lem3 44986	Lemma 3 for fmtno5 44988. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 3) = ;;;;;196608
Theorem	fmtno5lem4 44987	Lemma 4 for fmtno5 44988. (Contributed by AV, 30-Jul-2021.)
⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
Theorem	fmtno5 44988	The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
⊢ (FermatNo‘5) = ;;;;;;;;;4294967297
Theorem	fmtno0prm 44989	The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘0) ∈ ℙ
Theorem	fmtno1prm 44990	The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘1) ∈ ℙ
Theorem	fmtno2prm 44991	The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) ∈ ℙ
Theorem	257prm 44992	257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ ;;257 ∈ ℙ
Theorem	fmtno3prm 44993	The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ (FermatNo‘3) ∈ ℙ
Theorem	odz2prm2pw 44994	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 44995	Lemma for fmtnoprmfac1 44996: 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 44996*	Divisor of Fermat number (special form of Euler's result, see fmtnofac1 45001): 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 44997	Lemma for fmtnoprmfac2 44998. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)
Theorem	fmtnoprmfac2 44998*	Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 45000): 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 44999*	Lemma for fmtnofac2 45000 (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 45000*	Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 45001: 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))