Home Intuitionistic Logic ExplorerTheorem List (p. 35 of 137) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvn0m 3401 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)

Theoremn0rf 3402 An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class nonempty if and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3403 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0r 3403* An inhabited class is nonempty. See n0rf 3402 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremneq0r 3404* An inhabited class is nonempty. See n0rf 3402 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremreximdva0m 3405* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremn0mmoeu 3406* A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)

Theoremrex0 3407 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Theoremeq0 3408* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Theoremeqv 3409* The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Theoremnotm0 3410* A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)

Theoremnel0 3411* From the general negation of membership in , infer that is the empty set. (Contributed by BJ, 6-Oct-2018.)

Theorem0el 3412* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)

Theoremabvor0dc 3413* The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
DECID

Theoremabn0r 3414 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremabn0m 3415* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)

Theoremrabn0r 3416 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)

Theoremrabn0m 3417* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Theoremrab0 3418 Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrabeq0 3419 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Theoremabeq0 3420 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Theoremrabxmdc 3421* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
DECID

Theoremrabnc 3422* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Theoremun0 3423 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremin0 3424 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theorem0in 3425 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoreminv1 3426 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theoremunv 3427 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Theorem0ss 3428 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremss0b 3429 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)

Theoremss0 3430 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)

Theoremsseq0 3431 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssn0 3432 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Theoremabf 3433 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Theoremeq0rdv 3434* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Theoremcsbprc 3435 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)

Theoremun00 3436 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Theoremvss 3437 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj 3438* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Theoremdisjr 3439* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremdisj1 3440* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)

Theoremreldisj 3441 Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisj3 3442 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)

Theoremdisjne 3443 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjel 3444 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Theoremdisj2 3445 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Theoremssdisj 3446 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Theoremundisj1 3447 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Theoremundisj2 3448 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremssindif0im 3449 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Theoreminelcm 3450 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)

Theoremminel 3451 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)

Theoremundif4 3452 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdisjssun 3453 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssdif0im 3454 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)

Theoremvdif0im 3455 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifrab0eqim 3456* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoreminssdif0im 3457 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Theoremdifid 3458 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)

TheoremdifidALT 3459 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3458. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdif0 3460 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theorem0dif 3461 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Theoremdisjdif 3462 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Theoremdifin0 3463 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremundif1ss 3464 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundif2ss 3465 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremundifabs 3466 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Theoreminundifss 3467 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremdisjdif2 3468 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)

Theoremdifun2 3469 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Theoremundifss 3470 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremssdifin0 3471 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremssdifeq0 3472 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Theoremssundifim 3473 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremdifdifdirss 3474 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremuneqdifeqim 3475 Two ways that and can "partition" (when and don't overlap and is a part of ). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremr19.2m 3476* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1615). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)

Theoremr19.2mOLD 3477* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1615). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3476 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremr19.3rm 3478* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremr19.28m 3479* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.3rmv 3480* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremr19.9rmv 3481* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.28mv 3482* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremr19.45mv 3483* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.44mv 3484* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.27m 3485* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremr19.27mv 3486* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremrzal 3487* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremrexn0 3488* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3489). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremrexm 3489* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Theoremralidm 3490* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Theoremral0 3491 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)

Theoremrgenm 3492* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)

Theoremralf0 3493* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)

Theoremralm 3494 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaanlem 3495* Special case of raaan 3496 where is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)

Theoremraaan 3496* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Theoremraaanv 3497* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Theoremsbss 3498* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremsbcssg 3499 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)

Theoremdcun 3500 The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
DECID        DECID        DECID

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 13601-13601
 Copyright terms: Public domain < Previous  Next >