P. 108 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tskurn 10701	A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹:𝐴⟶𝑇) → ∪ ran 𝐹 ∈ 𝑇)
4.1.4  Grothendieck universes
Syntax	cgru 10702	Extend class notation to include the class of all Grothendieck universes.
class Univ
Definition	df-gru 10703*	A Grothendieck universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, Cartesian products etc. Grothendieck and alii, Séminaire de Géométrie Algébrique 4, Exposé I, p. 185. It was designed to give a precise meaning to the concepts of categories of sets, groups... (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
Theorem	elgrug 10704*	Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
Theorem	grutr 10705	A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝑈 ∈ Univ → Tr 𝑈)
Theorem	gruelss 10706	A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
Theorem	grupw 10707	A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)
Theorem	gruss 10708	Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈)
Theorem	grupr 10709	A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Theorem	gruurn 10710	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10711 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
Theorem	gruiun 10711*	If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruuni 10712	A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈)
Theorem	grurn 10713	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10711 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)
Theorem	gruima 10714	A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))
Theorem	gruel 10715	Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈)
Theorem	grusn 10716	A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)
Theorem	gruop 10717	A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Theorem	gruun 10718	A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)
Theorem	gruxp 10719	A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
Theorem	grumap 10720	A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑m 𝐵) ∈ 𝑈)
Theorem	gruixp 10721*	A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruiin 10722*	A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruf 10723	A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈)
Theorem	gruen 10724	A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈)
Theorem	gruwun 10725	A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)
Theorem	intgru 10726	The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)
Theorem	ingru 10727*	The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ))
Theorem	wfgru 10728	The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)
Theorem	grudomon 10729	Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → 𝐴 ∈ 𝑈)
Theorem	gruina 10730	If a Grothendieck universe 𝑈 is nonempty, then the height of the ordinals in 𝑈 is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
Theorem	grur1a 10731	A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
Theorem	grur1 10732	A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))
Theorem	grutsk1 10733	Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10695.) (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Theorem	grutsk 10734	Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥}
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
4.2.1  Introduce the Tarski-Grothendieck Axiom
Axiom	ax-groth 10735*	The Tarski-Grothendieck Axiom. For every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics". This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols (see grothprim 10746). An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
Theorem	axgroth5 10736*	The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
Theorem	axgroth2 10737*	Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
4.2.2  Derive the Power Set, Infinity and Choice Axioms
Theorem	grothpw 10738*	Derive the Axiom of Power Sets ax-pow 5300 from the Tarski-Grothendieck axiom ax-groth 10735. That it follows is mentioned by Bob Solovay at https://fomarchive.ugent.be/2008-March/012783.html 10735. Note that ax-pow 5300 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	grothpwex 10739	Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10735. Note that ax-pow 5300 is not used by the proof. Use axpweq 5286 to obtain ax-pow 5300. Use pwex 5315 or pwexg 5313 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
⊢ 𝒫 𝑥 ∈ V
Theorem	axgroth6 10740*	The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
Theorem	grothomex 10741	The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9553). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	grothac 10742	The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10380). This can be put in a more conventional form via ween 9946 and dfac8 10047. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see https://fomarchive.ugent.be/2008-March/012783.html 10047). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
⊢ dom card = V
Theorem	axgroth3 10743*	Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10346 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	axgroth4 10744*	Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10370 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑣 ∈ 𝑦 ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ (𝑦 ∩ 𝑣)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	grothprimlem 10745*	Lemma for grothprim 10746. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
Theorem	grothprim 10746*	The Tarski-Grothendieck Axiom ax-groth 10735 expanded into set theory primitives using 163 symbols (allowing the defined symbols ∧, ∨, ↔, and ∃). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧((𝑧 ∈ 𝑦 → ∃𝑣(𝑣 ∈ 𝑦 ∧ ∀𝑤(∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑧) → (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝑣)))) ∧ ∃𝑤((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (∀𝑣((𝑣 ∈ 𝑧 → ∃𝑡∀𝑢(∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑣 ∨ ℎ = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣 ∈ 𝑦 → (𝑣 ∈ 𝑧 ∨ ∃𝑢(𝑢 ∈ 𝑧 ∧ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))))) ∨ 𝑧 ∈ 𝑦))))
Theorem	grothtsk 10747	The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
⊢ ∪ Tarski = V
Theorem	inaprc 10748	An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ Inacc ∉ V
4.2.3  Tarski map function
Syntax	ctskm 10749	Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
Definition	df-tskm 10750*	A function that maps a set 𝑥 to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.)
⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
Theorem	tskmval 10751*	Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
Theorem	tskmid 10752	The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
Theorem	tskmcl 10753	A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (tarskiMap‘𝐴) ∈ Tarski
Theorem	sstskm 10754*	Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
Theorem	eltskm 10755*	Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
PART 5  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 11084). After that, we derive their basic properties, various operations like addition (df-add 11038) and sine (df-sin 16023), and subsets such as the integers (df-z 12514) and natural numbers (df-nn 12164).
5.1  Construction and axiomatization of real and complex numbers
5.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 10756	The set of positive integers, which is the set of natural numbers ω with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 11056. The actual set of Dedekind cuts is defined by df-np 10893.
class N
Syntax	cpli 10757	Positive integer addition.
class +N
Syntax	cmi 10758	Positive integer multiplication.
class ·N
Syntax	clti 10759	Positive integer ordering relation.
class <N
Syntax	cplpq 10760	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 10761	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 10762	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 10763	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 10764	Set of positive fractions.
class Q
Syntax	c1q 10765	The positive fraction constant 1.
class 1Q
Syntax	cerq 10766	Positive fraction equivalence class.
class [Q]
Syntax	cplq 10767	Positive fraction addition.
class +Q
Syntax	cmq 10768	Positive fraction multiplication.
class ·Q
Syntax	crq 10769	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 10770	Positive fraction ordering relation.
class <Q
Syntax	cnp 10771	Set of positive reals.
class P
Syntax	c1p 10772	Positive real constant 1.
class 1P
Syntax	cpp 10773	Positive real addition.
class +P
Syntax	cmp 10774	Positive real multiplication.
class ·P
Syntax	cltp 10775	Positive real ordering relation.
class <P
Syntax	cer 10776	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 10777	Set of signed reals.
class R
Syntax	c0r 10778	The signed real constant 0.
class 0R
Syntax	c1r 10779	The signed real constant 1.
class 1R
Syntax	cm1r 10780	The signed real constant -1.
class -1R
Syntax	cplr 10781	Signed real addition.
class +R
Syntax	cmr 10782	Signed real multiplication.
class ·R
Syntax	cltr 10783	Signed real ordering relation.
class <R
Definition	df-ni 10784	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11033, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N = (ω ∖ {∅})
Definition	df-pli 10785	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11033, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ +N = ( +o ↾ (N × N))
Definition	df-mi 10786	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11033, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ·N = ( ·o ↾ (N × N))
Definition	df-lti 10787	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11033, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.)
⊢ <N = ( E ∩ (N × N))
Theorem	elni 10788	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	elni2 10789	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	pinn 10790	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 10791	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 10792	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 10793	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N ∈ V
Theorem	0npi 10794	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ N
Theorem	1pi 10795	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
⊢ 1o ∈ N
Theorem	addpiord 10796	Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
Theorem	mulpiord 10797	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
Theorem	mulidpi 10798	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴)
Theorem	ltpiord 10799	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	ltsopi 10800	Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
⊢ <N Or N