Home Metamath Proof ExplorerTheorem List (p. 87 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtskpr 8601 If and are members of a Tarski's class, their unordered pair is also an element of the class. JFM CLASSES2 th. 3 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)

Theoremtskop 8602 If and are members of a Tarski's class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011.)

Theoremtskxpss 8603 A cross product of two parts of a Tarski's class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)

Theoremtskwe2 8604 A Tarski's class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)

Theoreminttsk 8605 The intersection of a collection of Tarski's classes is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoreminar1 8606 for a strongly inaccessible cardinal is equipotent to . (Contributed by Mario Carneiro, 6-Jun-2013.)

Theoremr1omALT 8607 The set of hereditarily finite sets is countable. This is a short proof as a consequence of inar1 8606, which requires AC. See r1om 8080 for a direct proof not requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrankcf 8608 Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of form a cofinal map into . (Contributed by Mario Carneiro, 27-May-2013.)

Theoreminatsk 8609 for a strongly inaccessible cardinal is a Tarski's class. (Contributed by Mario Carneiro, 8-Jun-2013.)

Theoremr1omtsk 8610 The set of hereditarily finite sets is a Tarski's class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.)

Theoremtskord 8611 A Tarski's class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)

Theoremtskcard 8612 An even more direct relationship than r1tskina 8613 to get an inacessible cardinal out of a Tarski's class: the size of any nonempty Tarski's class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremr1tskina 8613 There is a direct relationship between transitive Tarski's classes and inacessible cardinals: the Tarski's classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)

Theoremtskuni 8614 The union of an element of a transitive Tarski's class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremtskwun 8615 A nonempty transitive Tarski's class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremtskint 8616 The intersection of an element of a transitive Tarski's class is an element of the class. (Contributed by FL, 17-Apr-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremtskun 8617 The union of two elements of a transitive Tarski's class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.)

Theoremtskxp 8618 The cross product of two elements of a transitive Tarski's 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.)

Theoremtskmap 8619 Set exponentiation is an element of a transitive Tarski's class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoremtskurn 8620 A transitive Tarski's class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)

4.1.4  Grothendieck's universes

Syntaxcgru 8621 Extend class notation to include the class of all Grothendieck's universes.

Definitiondf-gru 8622* A Grothendieck's universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, cross 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.)

Theoremelgrug 8623* Properties of a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgrutr 8624 A Grothendieck's universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremgruelss 8625 A Grothendieck's universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgrupw 8626 A Grothendieck's universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruss 8627 Any subset of an element of a Grothendieck's universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgrupr 8628 A Grothendieck's universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruurn 8629 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8630 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruiun 8630* If is a family of elements of and the index set is an element of , then the indexed union is also an element of , where is a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruuni 8631 A Grothendieck's universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)

Theoremgrurn 8632 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8630 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruima 8633 A Grothendieck's universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruel 8634 Any element of an element of a Grothendieck's universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgrusn 8635 A Grothendieck's universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruop 8636 A Grothendieck's universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremgruun 8637 A Grothendieck's universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruxp 8638 A Grothendieck's universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgrumap 8639 A Grothendieck's universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruixp 8640* A Grothendieck's universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruiin 8641* A Grothendieck's universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruf 8642 A Grothendieck's universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremgruen 8643 A Grothendieck's universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremgruwun 8644 A nonempty Grothendieck's universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremintgru 8645 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Theoremingru 8646* The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremwfgru 8647 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)

Theoremgrudomon 8648 Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremgruina 8649 If a Grothendieck's universe is nonempty, then the height of the ordinals in is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)

Theoremgrur1a 8650 A characterization of Grothendieck's universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremgrur1 8651 A characterization of Grothendieck's universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremgrutsk1 8652 Grothendieck's universes are the same as transitive Tarski's classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 8614.) (Contributed by Mario Carneiro, 17-Jun-2013.)

Theoremgrutsk 8653 Grothendieck's universes are the same as transitive Tarski's classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.)

4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom

4.2.1  Introduce the Tarski-Grothendieck Axiom

Axiomax-groth 8654* The Tarski-Grothendieck Axiom. For every set there is an inaccessible cardinal such that is not in . 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 8665. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)

Theoremaxgroth5 8655* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)

Theoremaxgroth2 8656* Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)

4.2.2  Derive the Power Set, Infinity and Choice Axioms

Theoremgrothpw 8657* Derive the Axiom of Power Sets ax-pow 4337 from the Tarski-Grothendieck axiom ax-groth 8654. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4337 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.)

Theoremgrothpwex 8658 Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 8654. Note that ax-pow 4337 is not used by the proof. Use axpweq 4336 to obtain ax-pow 4337. (Contributed by Gérard Lang, 22-Jun-2009.)

Theoremaxgroth6 8659* The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set , there exists a set containing , the subsets of the members of , the power sets of the members of , and the subsets of of cardinality less than that of . (Contributed by NM, 21-Jun-2009.)

Theoremgrothomex 8660 The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 7554). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremgrothac 8661 The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8305). This can be put in a more conventional form via ween 7872 and dfac8 7971. 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). (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremaxgroth3 8662* Alternate version of the Tarski-Grothendieck Axiom. ax-cc 8271 is used to derive this version. (Contributed by NM, 26-Mar-2007.)

Theoremaxgroth4 8663* Alternate version of the Tarski-Grothendieck Axiom. ax-ac 8295 is used to derive this version. (Contributed by NM, 16-Apr-2007.)

Theoremgrothprimlem 8664* Lemma for grothprim 8665. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)

Theoremgrothprim 8665* The Tarski-Grothendieck Axiom ax-groth 8654 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.)

Theoremgrothtsk 8666 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)

Theoreminaprc 8667 An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)

4.2.3  Tarski map function

Syntaxctskm 8668 Extend class definition to include the map whose value is the smallest Tarski's class.

Definitiondf-tskm 8669* A function that maps a set to the smallest Tarski's class that contains the set. (Contributed by FL, 30-Dec-2010.)

Theoremtskmval 8670* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremtskmid 8671 The set is an element of the smallest Tarski's class that contains . CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremtskmcl 8672 A Tarski's class that contains is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremsstskm 8673* Being a part of . (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoremeltskm 8674* Belonging to . (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

PART 5  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers. We first construct and axiomitize real and complex numbers (e.g., ax-resscn 9003). After that we derive their basic properties, various operations like addition (df-add 8957) and sine (df-sin 12627), and subsets such as the integers (df-z 10239) and natural numbers (df-nn 9957).

5.1  Construction and axiomatization of real and complex numbers

5.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 8675 The set of positive integers, which is the set of natural numbers 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 8975. The actual set of Dedekind cuts is defined by df-np 8814.

Syntaxcmi 8677 Positive integer multiplication.

Syntaxclti 8678 Positive integer ordering relation.

Syntaxcmpq 8680 Positive pre-fraction multiplication.

Syntaxcltpq 8681 Positive pre-fraction ordering relation.

Syntaxceq 8682 Equivalence class used to construct positive fractions.

Syntaxcnq 8683 Set of positive fractions.

Syntaxc1q 8684 The positive fraction constant 1.

Syntaxcerq 8685 Positive fraction equivalence class.

Syntaxcmq 8687 Positive fraction multiplication.

Syntaxcrq 8688 Positive fraction reciprocal operation.

Syntaxcltq 8689 Positive fraction ordering relation.

Syntaxcnp 8690 Set of positive reals.

Syntaxc1p 8691 Positive real constant 1.

Syntaxcmp 8693 Positive real multiplication.

Syntaxcltp 8694 Positive real ordering relation.

Syntaxcmpr 8696 Signed real pre-multiplication.

Syntaxcer 8697 Equivalence class used to construct signed reals.

Syntaxcnr 8698 Set of signed reals.

Syntaxc0r 8699 The signed real constant 0.

Syntaxc1r 8700 The signed real constant 1.

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 32401-32447
 Copyright terms: Public domain < Previous  Next >