Home Intuitionistic Logic ExplorerTheorem List (p. 36 of 135) < 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 - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremifbieq1d 3501 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)

Theoremifbieq2i 3502 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq2d 3503 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremifbieq12i 3504 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Theoremifbieq12d 3505 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfifd 3506 Deduction version of nfif 3507. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremnfif 3507 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremifcldadc 3508 Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
DECID

Theoremifeq1dadc 3509 Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
DECID

Theoremifbothdadc 3510 A formula containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
DECID

Theoremifbothdc 3511 A wff containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
DECID

Theoremifiddc 3512 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
DECID

Theoremeqifdc 3513 Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
DECID

Theoremifcldcd 3514 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
DECID

Theoremifandc 3515 Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
DECID

Theoremifmdc 3516 If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
DECID

2.1.16  Power classes

Syntaxcpw 3517 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)

Theorempwjust 3518* Soundness justification theorem for df-pw 3519. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-pw 3519* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of . When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if is { 3 , 5 , 7 }, then is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)

Theorempweq 3520 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Theorempweqi 3521 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)

Theorempweqd 3522 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)

Theoremelpw 3523 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)

Theoremvelpw 3524* Setvar variable membership in a power class (common case). See elpw 3523. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelpwg 3525 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)

Theoremelpwi 3526 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Theoremelpwb 3527 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)

Theoremelpwid 3528 An element of a power class is a subclass. Deduction form of elpwi 3526. (Contributed by David Moews, 1-May-2017.)

Theoremelelpwi 3529 If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.)

Theoremnfpw 3530 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theorempwidg 3531 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theorempwid 3532 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)

Theorempwss 3533* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

2.1.17  Unordered and ordered pairs

Syntaxcsn 3534 Extend class notation to include singleton.

Syntaxcpr 3535 Extend class notation to include unordered pair.

Syntaxctp 3536 Extend class notation to include unordered triplet.

Syntaxcop 3537 Extend class notation to include ordered pair.

Syntaxcotp 3538 Extend class notation to include ordered triple.

Theoremsnjust 3539* Soundness justification theorem for df-sn 3540. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Definitiondf-sn 3540* 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 3548. (Contributed by NM, 5-Aug-1993.)

Definitiondf-pr 3541 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3609. For a more traditional definition, but requiring a dummy variable, see dfpr2 3553. (Contributed by NM, 5-Aug-1993.)

Definitiondf-tp 3542 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)

Definitiondf-op 3543* 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 3737 and opprc2 3738). For Kuratowski's actual definition when the arguments are sets, see dfop 3714.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3543 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3543 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 3544 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)

Theoremsneq 3545 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)

Theoremsneqi 3546 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremsneqd 3547 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)

Theoremdfsn2 3548 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelsng 3549 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 3550 There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Theoremvelsn 3551 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)

Theoremelsni 3552 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)

Theoremdfpr2 3553* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Theoremelprg 3554 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 3555 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 3556 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 3557 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 3558 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.)

Theoremprneli 3559 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremnelprd 3560 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Theoremeldifpr 3561 Membership in a set with two elements removed. Similar to eldifsn 3660 and eldiftp 3579. (Contributed by Mario Carneiro, 18-Jul-2017.)

Theoremrexdifpr 3562 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)

Theoremsnidg 3563 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Theoremsnidb 3564 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Theoremsnid 3565 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Theoremvsnid 3566 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelsn2g 3567 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 3568 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 3569* 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 3651 and snprc 3598. (Contributed by Jim Kingdon, 30-Aug-2018.)

Theoremralsnsg 3570* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremralsns 3571* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremrexsns 3572* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)

Theoremralsng 3573* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexsng 3574* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Theoremexsnrex 3575 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 3576* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 3577* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremeltpg 3578 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeldiftp 3579 Membership in a set with three elements removed. Similar to eldifsn 3660 and eldifpr 3561. (Contributed by David A. Wheeler, 22-Jul-2017.)

Theoremeltpi 3580 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 3581 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 3582* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 3583 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremralprg 3584* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 3585* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 3586* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 3587* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 3588* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 3589* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 3590* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 3591* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremsbcsng 3592* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 3593 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 3594 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremdisjsn 3595 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 3596 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 3597 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremsnprc 3598 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 3599* Special case of r19.12 2543 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)

Theoremrabsn 3600* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

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 13401-13495
 Copyright terms: Public domain < Previous  Next >