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Type | Label | Description |
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Statement | ||
Theorem | r1omALT 10201 | Alternate proof of r1om 9669, shorter as a consequence of inar1 10200, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅1‘ω) ≈ ω | ||
Theorem | rankcf 10202 | Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 𝐴 form a cofinal map into (rank‘𝐴). (Contributed by Mario Carneiro, 27-May-2013.) |
⊢ ¬ 𝐴 ≺ (cf‘(rank‘𝐴)) | ||
Theorem | inatsk 10203 | (𝑅1‘𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski) | ||
Theorem | r1omtsk 10204 | The set of hereditarily finite sets is a Tarski class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.) |
⊢ (𝑅1‘ω) ∈ Tarski | ||
Theorem | tskord 10205 | A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇) | ||
Theorem | tskcard 10206 | An even more direct relationship than r1tskina 10207 to get an inaccessible cardinal out of a Tarski class: the size of any nonempty Tarski class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc) | ||
Theorem | r1tskina 10207 | There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ (𝐴 ∈ On → ((𝑅1‘𝐴) ∈ Tarski ↔ (𝐴 = ∅ ∨ 𝐴 ∈ Inacc))) | ||
Theorem | tskuni 10208 | The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇) | ||
Theorem | tskwun 10209 | A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni) | ||
Theorem | tskint 10210 | The intersection of an element of a transitive Tarski class is an element of the class. (Contributed by FL, 17-Apr-2011.) (Revised by Mario Carneiro, 20-Sep-2014.) |
⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑇) | ||
Theorem | tskun 10211 | The union of two elements of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.) |
⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ∪ 𝐵) ∈ 𝑇) | ||
Theorem | tskxp 10212 | 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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ 𝑇) | ||
Theorem | tskmap 10213 | 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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ↑m 𝐵) ∈ 𝑇) | ||
Theorem | tskurn 10214 | 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) | ||
Syntax | cgru 10215 | Extend class notation to include the class of all Grothendieck universes. |
class Univ | ||
Definition | df-gru 10216* | 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 10217* | 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 10218 | A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝑈 ∈ Univ → Tr 𝑈) | ||
Theorem | gruelss 10219 | 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 10220 | 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 10221 | 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 10222 | 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 10223 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10224 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | ||
Theorem | gruiun 10224* | 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 10225 | A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | ||
Theorem | grurn 10226 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10224 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) | ||
Theorem | gruima 10227 | 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 10228 | 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 10229 | A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) | ||
Theorem | gruop 10230 | 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 10231 | 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 10232 | 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 10233 | 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 10234* | 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 10235* | 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 10236 | 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 10237 | 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 10238 | A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) | ||
Theorem | intgru 10239 | The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ) | ||
Theorem | ingru 10240* | 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 10241 | The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) | ||
Theorem | grudomon 10242 | 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 10243 | 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 10244 | A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈) | ||
Theorem | grur1 10245 | 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 10246 | Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10208.) (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ) | ||
Theorem | grutsk 10247 | 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} | ||
Axiom | ax-groth 10248* | 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 10259. 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 10249* | 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 10250* | 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))) | ||
Theorem | grothpw 10251* | Derive the Axiom of Power Sets ax-pow 5269 from the Tarski-Grothendieck axiom ax-groth 10248. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10248. Note that ax-pow 5269 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 10252 | Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10248. Note that ax-pow 5269 is not used by the proof. Use axpweq 5268 to obtain ax-pow 5269. Use pwex 5284 or pwexg 5282 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | axgroth6 10253* | 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 10254 | The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9109). 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 10255 | The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9894). This can be put in a more conventional form via ween 9464 and dfac8 9564. 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 9564). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
⊢ dom card = V | ||
Theorem | axgroth3 10256* | Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9860 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 10257* | Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9884 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 10258* | Lemma for grothprim 10259. 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 10259* | The Tarski-Grothendieck Axiom ax-groth 10248 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 10260 | The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.) |
⊢ ∪ Tarski = V | ||
Theorem | inaprc 10261 | An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ Inacc ∉ V | ||
Syntax | ctskm 10262 | Extend class definition to include the map whose value is the smallest Tarski class. |
class tarskiMap | ||
Definition | df-tskm 10263* | 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 10264* | 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 10265 | 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 10266 | 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 10267* | 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 10268* | 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))) | ||
This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 10597). After that, we derive their basic properties, various operations like addition (df-add 10551) and sine (df-sin 15426), and subsets such as the integers (df-z 11985) and natural numbers (df-nn 11642). | ||
Syntax | cnpi 10269 |
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 10569. The actual set of Dedekind cuts is defined by df-np 10406. |
class N | ||
Syntax | cpli 10270 | Positive integer addition. |
class +N | ||
Syntax | cmi 10271 | Positive integer multiplication. |
class ·N | ||
Syntax | clti 10272 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 10273 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 10274 | Positive pre-fraction multiplication. |
class ·pQ | ||
Syntax | cltpq 10275 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 10276 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 10277 | Set of positive fractions. |
class Q | ||
Syntax | c1q 10278 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cerq 10279 | Positive fraction equivalence class. |
class [Q] | ||
Syntax | cplq 10280 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 10281 | Positive fraction multiplication. |
class ·Q | ||
Syntax | crq 10282 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 10283 | Positive fraction ordering relation. |
class <Q | ||
Syntax | cnp 10284 | Set of positive reals. |
class P | ||
Syntax | c1p 10285 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 10286 | Positive real addition. |
class +P | ||
Syntax | cmp 10287 | Positive real multiplication. |
class ·P | ||
Syntax | cltp 10288 | Positive real ordering relation. |
class <P | ||
Syntax | cer 10289 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 10290 | Set of signed reals. |
class R | ||
Syntax | c0r 10291 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 10292 | The signed real constant 1. |
class 1R | ||
Syntax | cm1r 10293 | The signed real constant -1. |
class -1R | ||
Syntax | cplr 10294 | Signed real addition. |
class +R | ||
Syntax | cmr 10295 | Signed real multiplication. |
class ·R | ||
Syntax | cltr 10296 | Signed real ordering relation. |
class <R | ||
Definition | df-ni 10297 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10546, 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 10298 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10546, 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 10299 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10546, 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 10300 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10546, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.) |
⊢ <N = ( E ∩ (N × N)) |
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