Home | Intuitionistic Logic Explorer Theorem 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 |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ifbieq1d 3501 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
Theorem | ifbieq2i 3502 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq2d 3503 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3504 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3505 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3506 | Deduction version of nfif 3507. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3507 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifcldadc 3508 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
DECID | ||
Theorem | ifeq1dadc 3509 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | ifbothdadc 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 | ||
Theorem | ifbothdc 3511 | A wff containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifiddc 3512 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
DECID | ||
Theorem | eqifdc 3513 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
DECID | ||
Theorem | ifcldcd 3514 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifandc 3515 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
DECID | ||
Theorem | ifmdc 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 | ||
Syntax | cpw 3517 | Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.) |
Theorem | pwjust 3518* | Soundness justification theorem for df-pw 3519. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-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.) |
Theorem | pweq 3520 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3521 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3522 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3523 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | velpw 3524* | Setvar variable membership in a power class (common case). See elpw 3523. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3525 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3526 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwb 3527 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Theorem | elpwid 3528 | An element of a power class is a subclass. Deduction form of elpwi 3526. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3529 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3530 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3531 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3532 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3533* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3534 | Extend class notation to include singleton. |
Syntax | cpr 3535 | Extend class notation to include unordered pair. |
Syntax | ctp 3536 | Extend class notation to include unordered triplet. |
Syntax | cop 3537 | Extend class notation to include ordered pair. |
Syntax | cotp 3538 | Extend class notation to include ordered triple. |
Theorem | snjust 3539* | Soundness justification theorem for df-sn 3540. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-tp 3542 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-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.) |
Definition | df-ot 3544 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3545 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3546 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3547 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3548 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 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.) |
Theorem | elsn 3550 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3551 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3552 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3553* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 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.) |
Theorem | elpr 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.) |
Theorem | elpr2 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.) |
Theorem | elpri 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.) |
Theorem | nelpri 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.) |
Theorem | prneli 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.) |
Theorem | nelprd 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.) |
Theorem | eldifpr 3561 | Membership in a set with two elements removed. Similar to eldifsn 3660 and eldiftp 3579. (Contributed by Mario Carneiro, 18-Jul-2017.) |
Theorem | rexdifpr 3562 | Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
Theorem | snidg 3563 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.) |
Theorem | snidb 3564 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Theorem | snid 3565 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.) |
Theorem | vsnid 3566 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elsn2g 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.) |
Theorem | elsn2 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.) |
Theorem | mosn 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.) |
Theorem | ralsnsg 3570* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralsns 3571* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsns 3572* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
Theorem | ralsng 3573* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsng 3574* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
Theorem | exsnrex 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.) |
Theorem | ralsn 3576* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Theorem | rexsn 3577* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Theorem | eltpg 3578 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
Theorem | eldiftp 3579 | Membership in a set with three elements removed. Similar to eldifsn 3660 and eldifpr 3561. (Contributed by David A. Wheeler, 22-Jul-2017.) |
Theorem | eltpi 3580 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltp 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.) |
Theorem | dftp2 3582* | Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.) |
Theorem | nfpr 3583 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
Theorem | ralprg 3584* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexprg 3585* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | raltpg 3586* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextpg 3587* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralpr 3588* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexpr 3589* | Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | raltp 3590* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextp 3591* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | sbcsng 3592* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | nfsn 3593 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
Theorem | csbsng 3594 | Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | disjsn 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.) |
Theorem | disjsn2 3596 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
Theorem | disjpr2 3597 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
Theorem | snprc 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.) |
Theorem | r19.12sn 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.) |
Theorem | rabsn 3600* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |