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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ifbieq2d 3401 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ifbieq12i 3402 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
Theorem | ifbieq12d 3403 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | nfifd 3404 | Deduction version of nfif 3405. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | nfif 3405 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Theorem | ifcldadc 3406 | Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
DECID | ||
Theorem | ifeq1dadc 3407 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
DECID | ||
Theorem | ifbothdadc 3408 | 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 3409 | 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 3410 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
DECID | ||
Theorem | eqifdc 3411 | Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.) |
DECID | ||
Theorem | ifcldcd 3412 | Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
DECID | ||
Theorem | ifandc 3413 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
DECID | ||
Theorem | ifmdc 3414 | 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 3415 | 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 3416* | Soundness justification theorem for df-pw 3417. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-pw 3417* | 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 3418 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
Theorem | pweqi 3419 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | pweqd 3420 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Theorem | elpw 3421 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) |
Theorem | selpw 3422* | Setvar variable membership in a power class (common case). See elpw 3421. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elpwg 3423 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.) |
Theorem | elpwi 3424 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
Theorem | elpwid 3425 | An element of a power class is a subclass. Deduction form of elpwi 3424. (Contributed by David Moews, 1-May-2017.) |
Theorem | elelpwi 3426 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3427 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3428 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3429 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3430* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3431 | Extend class notation to include singleton. |
Syntax | cpr 3432 | Extend class notation to include unordered pair. |
Syntax | ctp 3433 | Extend class notation to include unordered triplet. |
Syntax | cop 3434 | Extend class notation to include ordered pair. |
Syntax | cotp 3435 | Extend class notation to include ordered triple. |
Theorem | snjust 3436* | Soundness justification theorem for df-sn 3437. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-sn 3437* | 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 3445. (Contributed by NM, 5-Aug-1993.) |
Definition | df-pr 3438 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so as proven by prcom 3503. For a more traditional definition, but requiring a dummy variable, see dfpr2 3450. (Contributed by NM, 5-Aug-1993.) |
Definition | df-tp 3439 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-op 3440* |
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 3629 and opprc2 3630). For
Kuratowski's actual definition when the arguments are sets, see dfop 3606.
Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as , which has different behavior from our df-op 3440 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3440 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 3441 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3442 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3443 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3444 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3445 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 3446 | 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 3447 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3448 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3449 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3450* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 3451 | 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 3452 | 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 3453 | 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 3454 | 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 3455 | 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 3456 | 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 3457 | 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 3458 | 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 3459 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Theorem | snid 3460 | 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 3461 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elsn2g 3462 | 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 3463 | 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 3464* | 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 3543 and snprc 3492. (Contributed by Jim Kingdon, 30-Aug-2018.) |
Theorem | ralsnsg 3465* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralsns 3466* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsns 3467* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
Theorem | ralsng 3468* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsng 3469* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
Theorem | exsnrex 3470 | 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 3471* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Theorem | rexsn 3472* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Theorem | eltpg 3473 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltpi 3474 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltp 3475 | 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 3476* | 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 3477 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
Theorem | ralprg 3478* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexprg 3479* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | raltpg 3480* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextpg 3481* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralpr 3482* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexpr 3483* | 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 3484* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextp 3485* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | sbcsng 3486* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | nfsn 3487 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
Theorem | csbsng 3488 | Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | disjsn 3489 | 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 3490 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
Theorem | disjpr2 3491 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
Theorem | snprc 3492 | 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 3493* | Special case of r19.12 2474 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 3494* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
Theorem | rabrsndc 3495* | 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 3496* | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | euabsn 3497 | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
Theorem | reusn 3498* | 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 3499 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
Theorem | rabsneu 3500 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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