Home Metamath Proof ExplorerTheorem List (p. 103 of 449) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28687) Hilbert Space Explorer (28688-30210) Users' Mathboxes (30211-44886)

Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtskxp 10201 The Cartesian product of two elements of a transitive Tarski class is an element of the class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ∈ 𝑇)

Theoremtskmap 10202 Set exponentiation is an element of a transitive Tarski class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐵𝑇) → (𝐴m 𝐵) ∈ 𝑇)

Theoremtskurn 10203 A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)

4.1.4  Grothendieck universes

Syntaxcgru 10204 Extend class notation to include the class of all Grothendieck universes.
class Univ

Definitiondf-gru 10205* 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 10206* 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 10207 A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ Univ → Tr 𝑈)

Theoremgruelss 10208 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 10209 A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)

Theoremgruss 10210 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 10211 A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Theoremgruurn 10212 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10213 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Theoremgruiun 10213* 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 10214 A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)

Theoremgrurn 10215 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10213 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Theoremgruima 10216 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 10217 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 10218 A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)

Theoremgruop 10219 A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Theoremgruun 10220 A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)

Theoremgruxp 10221 A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)

Theoremgrumap 10222 A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴m 𝐵) ∈ 𝑈)

Theoremgruixp 10223* 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 10224* A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)

Theoremgruf 10225 A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)

Theoremgruen 10226 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 10227 A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)

Theoremintgru 10228 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)

Theoremingru 10229* 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 10230 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
(𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)

Theoremgrudomon 10231 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 10232 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 10233 A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 = (𝑈 ∩ On)       (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)

Theoremgrur1 10234 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 10235 Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10197.) (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)

Theoremgrutsk 10236 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 10237* 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 10248. 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 10238* 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 10239* 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 10240* Derive the Axiom of Power Sets ax-pow 5257 from the Tarski-Grothendieck axiom ax-groth 10237. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10237. Note that ax-pow 5257 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremgrothpwex 10241 Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10237. Note that ax-pow 5257 is not used by the proof. Use axpweq 5256 to obtain ax-pow 5257. Use pwex 5272 or pwexg 5270 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝒫 𝑥 ∈ V

Theoremaxgroth6 10242* 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 10243 The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9098). 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 10244 The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9883). This can be put in a more conventional form via ween 9453 and dfac8 9553. 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 9553). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
dom card = V

Theoremaxgroth3 10245* Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9849 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))

Theoremaxgroth4 10246* Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9873 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))

Theoremgrothprimlem 10247* Lemma for grothprim 10248. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))

Theoremgrothprim 10248* The Tarski-Grothendieck Axiom ax-groth 10237 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 10249 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Tarski = V

Theoreminaprc 10250 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 10251 Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap

Definitiondf-tskm 10252* 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 10253* 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 10254 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 10255 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 10256* 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 10257* 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 10586). After that, we derive their basic properties, various operations like addition (df-add 10540) and sine (df-sin 15415), and subsets such as the integers (df-z 11974) and natural numbers (df-nn 11631).

5.1  Construction and axiomatization of real and complex numbers

5.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 10258 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 10558. The actual set of Dedekind cuts is defined by df-np 10395.

class N

class +N

Syntaxcmi 10260 Positive integer multiplication.
class ·N

Syntaxclti 10261 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 10263 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 10264 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 10265 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 10266 Set of positive fractions.
class Q

Syntaxc1q 10267 The positive fraction constant 1.
class 1Q

Syntaxcerq 10268 Positive fraction equivalence class.
class [Q]

class +Q

Syntaxcmq 10270 Positive fraction multiplication.
class ·Q

Syntaxcrq 10271 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 10272 Positive fraction ordering relation.
class <Q

Syntaxcnp 10273 Set of positive reals.
class P

Syntaxc1p 10274 Positive real constant 1.
class 1P

class +P

Syntaxcmp 10276 Positive real multiplication.
class ·P

Syntaxcltp 10277 Positive real ordering relation.
class <P

Syntaxcer 10278 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 10279 Set of signed reals.
class R

Syntaxc0r 10280 The signed real constant 0.
class 0R

Syntaxc1r 10281 The signed real constant 1.
class 1R

Syntaxcm1r 10282 The signed real constant -1.
class -1R

class +R

Syntaxcmr 10284 Signed real multiplication.
class ·R

Syntaxcltr 10285 Signed real ordering relation.
class <R

Definitiondf-ni 10286 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10535, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
N = (ω ∖ {∅})

Definitiondf-pli 10287 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10535, 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 10288 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10535, 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 10289 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10535, 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 10290 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Theoremelni2 10291 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))

Theorempinn 10292 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
(𝐴N𝐴 ∈ ω)

Theorempion 10293 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
(𝐴N𝐴 ∈ On)

Theorempiord 10294 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
(𝐴N → Ord 𝐴)

Theoremniex 10295 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
N ∈ V

Theorem0npi 10296 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
¬ ∅ ∈ N

Theorem1pi 10297 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
1oN

Theoremaddpiord 10298 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 10299 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 10300 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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44886
 Copyright terms: Public domain < Previous  Next >