Home Intuitionistic Logic ExplorerTheorem List (p. 35 of 110) < 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

Theoremelelpwi 3401 If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.)

Theoremnfpw 3402 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theorempwidg 3403 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theorempwid 3404 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)

Theorempwss 3405* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

2.1.17  Unordered and ordered pairs

Syntaxcsn 3406 Extend class notation to include singleton.

Syntaxcpr 3407 Extend class notation to include unordered pair.

Syntaxctp 3408 Extend class notation to include unordered triplet.

Syntaxcop 3409 Extend class notation to include ordered pair.

Syntaxcotp 3410 Extend class notation to include ordered triple.

Theoremsnjust 3411* Soundness justification theorem for df-sn 3412. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-sn 3412* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of , although it is not very meaningful in this case. For an alternate definition see dfsn2 3420. (Contributed by NM, 5-Aug-1993.)

Definitiondf-pr 3413 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3476. For a more traditional definition, but requiring a dummy variable, see dfpr2 3425. (Contributed by NM, 5-Aug-1993.)

Definitiondf-tp 3414 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)

Definitiondf-op 3415* Definition of an ordered pair, equivalent to Kuratowski's definition when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3600 and opprc2 3601). For Kuratowski's actual definition when the arguments are sets, see dfop 3577.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3415 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3415 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition _2 . This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is _3 , but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Definitiondf-ot 3416 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)

Theoremsneq 3417 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremsneqi 3418 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremsneqd 3419 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremdfsn2 3420 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelsng 3421 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremelsn 3422 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremvelsn 3423 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)

Theoremelsni 3424 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)

Theoremdfpr2 3425* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelprg 3426 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)

Theoremelpr 3427 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremelpr2 3428 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)

Theoremelpri 3429 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)

Theoremnelpri 3430 If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)

Theoremsnidg 3431 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Theoremsnidb 3432 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Theoremsnid 3433 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Theoremvsnid 3434 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelsn2g 3435 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsn2 3436 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 12-Jun-1994.)

Theoremmosn 3437* A singleton has at most one element. This works whether is a proper class or not, and in that sense can be seen as encompassing both snmg 3516 and snprc 3465. (Contributed by Jim Kingdon, 30-Aug-2018.)

Theoremralsnsg 3438* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremralsns 3439* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremrexsns 3440* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)

Theoremralsng 3441* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexsng 3442* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Theoremexsnrex 3443 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theoremralsn 3444* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 3445* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremeltpg 3446 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeltpi 3447 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 3448 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdftp2 3449* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 3450 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremralprg 3451* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 3452* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 3453* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 3454* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 3455* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 3456* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 3457* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 3458* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremsbcsng 3459* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 3460 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 3461 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremdisjsn 3462 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Theoremdisjsn2 3463 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 3464 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremsnprc 3465 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

Theoremr19.12sn 3466* Special case of r19.12 2467 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)

Theoremrabsn 3467* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Theoremrabrsndc 3468* A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
DECID

Theoremeuabsn2 3469* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremeuabsn 3470 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Theoremreusn 3471* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremabsneu 3472 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Theoremrabsneu 3473 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremeusn 3474* Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.)

Theoremrabsnt 3475* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremprcom 3476 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq1 3477 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Theorempreq2 3478 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq12 3479 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1i 3480 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2i 3481 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12i 3482 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1d 3483 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2d 3484 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12d 3485 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theoremtpeq1 3486 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq2 3487 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq3 3488 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq1d 3489 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq2d 3490 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq3d 3491 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq123d 3492 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtprot 3493 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpcoma 3494 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpcomb 3495 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpass 3496 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdass 3497 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdassr 3498 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremtpidm12 3499 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm13 3500 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

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 10901-10953
 Copyright terms: Public domain < Previous  Next >