Home Metamath Proof ExplorerTheorem List (p. 41 of 321) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22283) Hilbert Space Explorer (22284-23806) Users' Mathboxes (23807-32095)

Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunissb 4001* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Theoremuniss2 4002* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4091 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)

Theoremunidif 4003* If the difference contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.)

Theoremssunieq 4004* Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Theoremunimax 4005* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)

2.1.19  The intersection of a class

Syntaxcint 4006 Extend class notation to include the intersection of a class (read: 'intersect ').

Definitiondf-int 4007* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, . Compare this with the intersection of two classes, df-in 3284. (Contributed by NM, 18-Aug-1993.)

Theoremdfint2 4008* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Theoreminteq 4009 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Theoreminteqi 4010 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoreminteqd 4011 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoremelint 4012* Membership in class intersection. (Contributed by NM, 21-May-1994.)

Theoremelint2 4013* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Theoremelintg 4014* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Theoremelinti 4015 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremnfint 4016 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremelintab 4017* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Theoremelintrab 4018* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Theoremelintrabg 4019* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Theoremint0 4020 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Theoremintss1 4021 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)

Theoremssint 4022* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Theoremssintab 4023* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremssintub 4024* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Theoremssmin 4025* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Theoremintmin 4026* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremintss 4027 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Theoremintssuni 4028 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Theoremssintrab 4029* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)

Theoremunissint 4030 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4043). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremintssuni2 4031 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)

Theoremintminss 4032* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Theoremintmin2 4033* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Theoremintmin3 4034* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)

Theoremintmin4 4035* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Theoremintab 4036* The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . Typically, abrexex2 5954 or abexssex 5955 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremint0el 4037 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Theoremintun 4038 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Theoremintpr 4039 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Theoremintprg 4040 The intersection of a pair is the intersection of its members. Closed form of intpr 4039. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Theoremintsng 4041 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremintsn 4042 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Theoremuniintsn 4043* Two ways to express " is a singleton." See also en1 7124, en1b 7125, card1 7802, and eusn 3837. (Contributed by NM, 2-Aug-2010.)

Theoremuniintab 4044 The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremintunsn 4045 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Theoremrint0 4046 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint 4047* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint2 4048* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

2.1.20  Indexed union and intersection

Syntaxciun 4049 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same union symbol as cuni 3971. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Syntaxciin 4050 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same intersection symbol as cint 4006. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Definitiondf-iun 4051* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, is independent of (although this is not required by the definition), and depends on i.e. can be read informally as . We call the index, the index set, and the indexed set. In most books, is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that and are in the same distinct variable group (meaning cannot depend on ) and that and do not share a distinct variable group (meaning that can be thought of as i.e. can be substituted with a class expression containing ). An alternate definition tying indexed union to ordinary union is dfiun2 4081. Theorem uniiun 4099 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5947 and funiunfv 5948 are useful when is a function. (Contributed by NM, 27-Jun-1998.)

Definitiondf-iin 4052* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4051. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4082. Theorem intiin 4100 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Theoremeliun 4053* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)

Theoremeliin 4054* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Theoremiuncom 4055* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 4056 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconst 4057* Indexed union of a constant class, i.e. where does not depend on . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinconst 4058* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremiuniin 4059* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiunss1 4060* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 4061* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 4062* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 4063* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 4064 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2 4065 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2 4066 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2i 4067 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2i 4068 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2d 4069 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Theoremiuneq2dv 4070* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)

Theoremiineq2dv 4071* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Theoremiuneq1d 4072* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq12d 4073* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq2d 4074* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Theoremnfiun 4075 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiin 4076 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiu1 4077 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Theoremnfii1 4078 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremdfiun2g 4079* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiin2g 4080* Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Theoremdfiun2 4081* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdfiin2 4082* Alternate definition of indexed intersection when is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremcbviun 4083* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Theoremcbviin 4084* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbviunv 4085* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Theoremcbviinv 4086* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Theoremiunss 4087* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun 4088* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2 4089 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2s 4090* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Theoremiunss2 4091* A subclass condition on the members of two indexed classes and that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4002. (Contributed by NM, 9-Dec-2004.)

Theoremiunab 4092* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Theoremiunrab 4093* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremiunxdif2 4094* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Theoremssiinf 4095 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Theoremssiin 4096* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremiinss 4097* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss2 4098 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Theoremuniiun 4099* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Theoremintiin 4100* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

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 32001-32095
 Copyright terms: Public domain < Previous  Next >