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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ifcldadc 3501 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
DECID | ||
Theorem | ifeq1dadc 3502 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | ifbothdadc 3503 | 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 3504 | 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 3505 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
DECID | ||
Theorem | eqifdc 3506 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
DECID | ||
Theorem | ifcldcd 3507 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifandc 3508 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
DECID | ||
Theorem | ifmdc 3509 | 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 3510 | 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 3511* | Soundness justification theorem for df-pw 3512. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-pw 3512* | 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 3513 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3514 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3515 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3516 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | velpw 3517* | Setvar variable membership in a power class (common case). See elpw 3516. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3518 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3519 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwb 3520 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Theorem | elpwid 3521 | An element of a power class is a subclass. Deduction form of elpwi 3519. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3522 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3523 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3524 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3525 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3526* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3527 | Extend class notation to include singleton. |
Syntax | cpr 3528 | Extend class notation to include unordered pair. |
Syntax | ctp 3529 | Extend class notation to include unordered triplet. |
Syntax | cop 3530 | Extend class notation to include ordered pair. |
Syntax | cotp 3531 | Extend class notation to include ordered triple. |
Theorem | snjust 3532* | Soundness justification theorem for df-sn 3533. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-sn 3533* | 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 3541. (Contributed by NM, 5-Aug-1993.) |
Definition | df-pr 3534 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3599. For a more traditional definition, but requiring a dummy variable, see dfpr2 3546. (Contributed by NM, 5-Aug-1993.) |
Definition | df-tp 3535 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-op 3536* |
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 3727 and opprc2 3728). For
Kuratowski's actual definition when the arguments are sets, see dfop 3704.
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3536 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3536 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 3537 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3538 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3539 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3540 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3541 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 3542 | 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 3543 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3544 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3545 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3546* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 3547 | 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 3548 | 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 3549 | 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 3550 | 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 3551 | 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 3552 | 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 3553 | 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 | snidg 3554 | 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 3555 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Theorem | snid 3556 | 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 3557 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elsn2g 3558 | 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 3559 | 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 3560* | 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 3641 and snprc 3588. (Contributed by Jim Kingdon, 30-Aug-2018.) |
Theorem | ralsnsg 3561* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralsns 3562* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsns 3563* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
Theorem | ralsng 3564* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsng 3565* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
Theorem | exsnrex 3566 | 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 3567* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Theorem | rexsn 3568* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Theorem | eltpg 3569 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltpi 3570 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltp 3571 | 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 3572* | 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 3573 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
Theorem | ralprg 3574* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexprg 3575* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | raltpg 3576* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextpg 3577* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralpr 3578* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexpr 3579* | 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 3580* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextp 3581* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | sbcsng 3582* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | nfsn 3583 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
Theorem | csbsng 3584 | Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | disjsn 3585 | 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 3586 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
Theorem | disjpr2 3587 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
Theorem | snprc 3588 | 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 3589* | Special case of r19.12 2538 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 3590* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
Theorem | rabrsndc 3591* | 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 | ||
Theorem | euabsn2 3592* | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | euabsn 3593 | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
Theorem | reusn 3594* | 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.) |
Theorem | absneu 3595 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
Theorem | rabsneu 3596 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | eusn 3597* | Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.) |
Theorem | rabsnt 3598* | 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.) |
Theorem | prcom 3599 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Theorem | preq1 3600 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
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