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Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
4.1.4  Grothendieck universes
 
Syntaxcgru 10201 Extend class notation to include the class of all Grothendieck universes.
class Univ
 
Definitiondf-gru 10202* 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 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢))}
 
Theoremelgrug 10203* Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
(𝑈𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
 
Theoremgrutr 10204 A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ Univ → Tr 𝑈)
 
Theoremgruelss 10205 A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
 
Theoremgrupw 10206 A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
 
Theoremgruss 10207 Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
 
Theoremgrupr 10208 A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremgruurn 10209 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10210 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
 
Theoremgruiun 10210* If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union 𝑥𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
 
Theoremgruuni 10211 A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
 
Theoremgrurn 10212 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10210 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
 
Theoremgruima 10213 A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))
 
Theoremgruel 10214 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 ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
 
Theoremgrusn 10215 A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
 
Theoremgruop 10216 A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 
Theoremgruun 10217 A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
 
Theoremgruxp 10218 A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
 
Theoremgrumap 10219 A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴m 𝐵) ∈ 𝑈)
 
Theoremgruixp 10220* A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → X𝑥𝐴 𝐵𝑈)
 
Theoremgruiin 10221* A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
 
Theoremgruf 10222 A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)
 
Theoremgruen 10223 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 ∧ 𝐴𝑈 ∧ (𝐵𝑈𝐵𝐴)) → 𝐴𝑈)
 
Theoremgruwun 10224 A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)
 
Theoremintgru 10225 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
 
Theoremingru 10226* 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 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈 ∈ Univ → (𝑈𝐴) ∈ Univ))
 
Theoremwfgru 10227 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
(𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)
 
Theoremgrudomon 10228 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 ∧ (𝐵𝑈𝐴𝐵)) → 𝐴𝑈)
 
Theoremgruina 10229 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)
 
Theoremgrur1a 10230 A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 = (𝑈 ∩ On)       (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)
 
Theoremgrur1 10231 A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 = (𝑈 ∩ On)       ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1𝐴))
 
Theoremgrutsk1 10232 Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10194.) (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
 
Theoremgrutsk 10233 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
 
Axiomax-groth 10234* 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 10245. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
 
Theoremaxgroth5 10235* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
 
Theoremaxgroth2 10236* Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑦𝑧𝑧𝑦)))
 
4.2.2  Derive the Power Set, Infinity and Choice Axioms
 
Theoremgrothpw 10237* Derive the Axiom of Power Sets ax-pow 5258 from the Tarski-Grothendieck axiom ax-groth 10234. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10234. Note that ax-pow 5258 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
 
Theoremgrothpwex 10238 Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10234. Note that ax-pow 5258 is not used by the proof. Use axpweq 5257 to obtain ax-pow 5258. Use pwex 5273 or pwexg 5271 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝒫 𝑥 ∈ V
 
Theoremaxgroth6 10239* 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(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
 
Theoremgrothomex 10240 The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9095). 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
 
Theoremgrothac 10241 The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9880). This can be put in a more conventional form via ween 9450 and dfac8 9550. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 9550). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
dom card = V
 
Theoremaxgroth3 10242* Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9846 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
 
Theoremaxgroth4 10243* Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9870 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))
 
Theoremgrothprimlem 10244* Lemma for grothprim 10245. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
 
Theoremgrothprim 10245* The Tarski-Grothendieck Axiom ax-groth 10234 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(𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))
 
Theoremgrothtsk 10246 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Tarski = V
 
Theoreminaprc 10247 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
 
Syntaxctskm 10248 Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
 
Definitiondf-tskm 10249* 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 ∣ 𝑥𝑦})
 
Theoremtskmval 10250* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
(𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
 
Theoremtskmid 10251 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.)
(𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
 
Theoremtskmcl 10252 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
 
Theoremsstskm 10253* Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
(𝐴𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
 
Theoremeltskm 10254* Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
(𝐴𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐵𝑥)))
 
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 10583). After that, we derive their basic properties, various operations like addition (df-add 10537) and sine (df-sin 15413), and subsets such as the integers (df-z 11971) and natural numbers (df-nn 11628).

 
5.1  Construction and axiomatization of real and complex numbers
 
5.1.1  Dedekind-cut construction of real and complex numbers
 
Syntaxcnpi 10255 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 10555. The actual set of Dedekind cuts is defined by df-np 10392.

class N
 
Syntaxcpli 10256 Positive integer addition.
class +N
 
Syntaxcmi 10257 Positive integer multiplication.
class ·N
 
Syntaxclti 10258 Positive integer ordering relation.
class <N
 
Syntaxcplpq 10259 Positive pre-fraction addition.
class +pQ
 
Syntaxcmpq 10260 Positive pre-fraction multiplication.
class ·pQ
 
Syntaxcltpq 10261 Positive pre-fraction ordering relation.
class <pQ
 
Syntaxceq 10262 Equivalence class used to construct positive fractions.
class ~Q
 
Syntaxcnq 10263 Set of positive fractions.
class Q
 
Syntaxc1q 10264 The positive fraction constant 1.
class 1Q
 
Syntaxcerq 10265 Positive fraction equivalence class.
class [Q]
 
Syntaxcplq 10266 Positive fraction addition.
class +Q
 
Syntaxcmq 10267 Positive fraction multiplication.
class ·Q
 
Syntaxcrq 10268 Positive fraction reciprocal operation.
class *Q
 
Syntaxcltq 10269 Positive fraction ordering relation.
class <Q
 
Syntaxcnp 10270 Set of positive reals.
class P
 
Syntaxc1p 10271 Positive real constant 1.
class 1P
 
Syntaxcpp 10272 Positive real addition.
class +P
 
Syntaxcmp 10273 Positive real multiplication.
class ·P
 
Syntaxcltp 10274 Positive real ordering relation.
class <P
 
Syntaxcer 10275 Equivalence class used to construct signed reals.
class ~R
 
Syntaxcnr 10276 Set of signed reals.
class R
 
Syntaxc0r 10277 The signed real constant 0.
class 0R
 
Syntaxc1r 10278 The signed real constant 1.
class 1R
 
Syntaxcm1r 10279 The signed real constant -1.
class -1R
 
Syntaxcplr 10280 Signed real addition.
class +R
 
Syntaxcmr 10281 Signed real multiplication.
class ·R
 
Syntaxcltr 10282 Signed real ordering relation.
class <R
 
Definitiondf-ni 10283 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
N = (ω ∖ {∅})
 
Definitiondf-pli 10284 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
+N = ( +o ↾ (N × N))
 
Definitiondf-mi 10285 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
·N = ( ·o ↾ (N × N))
 
Definitiondf-lti 10286 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.)
<N = ( E ∩ (N × N))
 
Theoremelni 10287 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
 
Theoremelni2 10288 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
 
Theorempinn 10289 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
(𝐴N𝐴 ∈ ω)
 
Theorempion 10290 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
(𝐴N𝐴 ∈ On)
 
Theorempiord 10291 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
(𝐴N → Ord 𝐴)
 
Theoremniex 10292 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
N ∈ V
 
Theorem0npi 10293 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
¬ ∅ ∈ N
 
Theorem1pi 10294 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
1oN
 
Theoremaddpiord 10295 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
 
Theoremmulpiord 10296 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
 
Theoremmulidpi 10297 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.)
(𝐴N → (𝐴 ·N 1o) = 𝐴)
 
Theoremltpiord 10298 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.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
 
Theoremltsopi 10299 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
 
Theoremltrelpi 10300 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)
<N ⊆ (N × N)
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