Home Metamath Proof ExplorerTheorem List (p. 40 of 315) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21459) Hilbert Space Explorer (21460-22982) Users' Mathboxes (22983-31404)

Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiineq2dv 3901* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Theoremiuneq1d 3902* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq12d 3903* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq2d 3904* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Theoremnfiun 3905 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiin 3906 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiu1 3907 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Theoremnfii1 3908 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremdfiun2g 3909* 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 3910* Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Theoremdfiun2 3911* 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 3912* 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 3913* 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 3914* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbviunv 3915* 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 3916* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Theoremiunss 3917* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun 3918* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2 3919 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2s 3920* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Theoremiunss2 3921* 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 3832. (Contributed by NM, 9-Dec-2004.)

Theoremiunab 3922* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Theoremiunrab 3923* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremiunxdif2 3924* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Theoremssiinf 3925 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Theoremssiin 3926* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremiinss 3927* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss2 3928 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Theoremuniiun 3929* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Theoremintiin 3930* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Theoremiunid 3931* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Theoremiun0 3932 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iun 3933 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iin 3934 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Theoremviin 3935* Indexed intersection with a universal index class. When doesn't depend on , this evaluates to by 19.3 1760 and abid2 2375. When , this evaluates to by intiin 3930 and intv 4158. (Contributed by NM, 11-Sep-2008.)

Theoremiunn0 3936* There is a non-empty class in an indexed collection iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinab 3937* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab 3938* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab2 3939* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiunin2 3940* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3929 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Theoremiunin1 3941* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3929 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremiinun2 3942* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theoremiundif2 3943* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3930 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theorem2iunin 3944* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Theoremiindif2 3945* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3929 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Theoremiinin2 3946* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremiinin1 3947* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremelriin 3948* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremriin0 3949* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinn0 3950* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinrab 3951* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremiinxsng 3952* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiinxprg 3953* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Theoremiunxsng 3954* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Theoremiunxsn 3955* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)

Theoremiunun 3956 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxun 3957 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxiun 3958* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremiinuni 3959* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiununi 3960* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremsspwuni 3961 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theorempwssb 3962* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Theoremelpwuni 3963 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theoremiinpw 3964* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Theoremiunpwss 3965* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)

Theoremrintn0 3966 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)

2.1.21  Disjointness

Syntaxwdisj 3967 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj

Definitiondf-disj 3968* A collection of classes is disjoint when for each element , it is in for at most one . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
Disj

Theoremdfdisj2 3969* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Disj

Theoremdisjss2 3970 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2 3971 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2dv 3972* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjss1 3973* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1 3974* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1d 3975* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq12d 3976* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisj 3977* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisjv 3978* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Disj Disj

Theoremnfdisj 3979 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremnfdisj1 3980 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjor 3981* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

TheoremdisjmoOLD 3982* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)

Theoremdisjors 3983* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji2 3984* Property of a disjoint collection: if and , and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji 3985* Property of a disjoint collection: if and have a common element , then . (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoreminvdisj 3986* If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremdisjiun 3987* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

TheoremdisjiunOLD 3988* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)

Theoremsndisj 3989 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theorem0disj 3990 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxsn 3991* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjx0 3992 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjprg 3993* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxiun 3994* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that and may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj Disj

Theoremdisjxun 3995* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj

Theoremdisjss3 3996* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Disj Disj

2.1.22  Binary relations

Syntaxwbr 3997 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 9006.)

Definitiondf-br 3998 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class often denotes a relation such as " " that compares two classes and , which might be numbers such as and (see df-ltxr 8840 for the specific definition of ). As a wff, relations are true or false. For example, (ex-br 20762). Often class meets the criteria to be defined in df-rel 4676, and in particular may be a function (see df-fun 4683). This definition of relations is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when is a proper class. (Contributed by NM, 31-Dec-1993.)

Theorembreq 3999 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Theorembreq1 4000 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

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 31401-31404
 Copyright terms: Public domain < Previous  Next >