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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elelpwi 3401 | If belongs to a part of then belongs to . (Contributed by FL, 3-Aug-2009.) |
Theorem | nfpw 3402 | Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | pwidg 3403 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Theorem | pwid 3404 | A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) |
Theorem | pwss 3405* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
Syntax | csn 3406 | Extend class notation to include singleton. |
Syntax | cpr 3407 | Extend class notation to include unordered pair. |
Syntax | ctp 3408 | Extend class notation to include unordered triplet. |
Syntax | cop 3409 | Extend class notation to include ordered pair. |
Syntax | cotp 3410 | Extend class notation to include ordered triple. |
Theorem | snjust 3411* | Soundness justification theorem for df-sn 3412. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-tp 3414 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
Definition | df-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.) |
Definition | df-ot 3416 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
Theorem | sneq 3417 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
Theorem | sneqi 3418 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | sneqd 3419 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
Theorem | dfsn2 3420 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elsng 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.) |
Theorem | elsn 3422 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Theorem | velsn 3423 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
Theorem | elsni 3424 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
Theorem | dfpr2 3425* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Theorem | elprg 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.) |
Theorem | elpr 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.) |
Theorem | elpr2 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.) |
Theorem | elpri 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.) |
Theorem | nelpri 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.) |
Theorem | snidg 3431 | 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 3432 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Theorem | snid 3433 | 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 3434 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elsn2g 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.) |
Theorem | elsn2 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.) |
Theorem | mosn 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.) |
Theorem | ralsnsg 3438* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralsns 3439* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsns 3440* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
Theorem | ralsng 3441* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexsng 3442* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
Theorem | exsnrex 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.) |
Theorem | ralsn 3444* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
Theorem | rexsn 3445* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Theorem | eltpg 3446 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltpi 3447 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Theorem | eltp 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.) |
Theorem | dftp2 3449* | 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 3450 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
Theorem | ralprg 3451* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexprg 3452* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | raltpg 3453* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextpg 3454* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | ralpr 3455* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rexpr 3456* | 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 3457* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | rextp 3458* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | sbcsng 3459* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Theorem | nfsn 3460 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
Theorem | csbsng 3461 | Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | disjsn 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.) |
Theorem | disjsn2 3463 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
Theorem | disjpr2 3464 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
Theorem | snprc 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.) |
Theorem | r19.12sn 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.) |
Theorem | rabsn 3467* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
Theorem | rabrsndc 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 | ||
Theorem | euabsn2 3469* | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | euabsn 3470 | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
Theorem | reusn 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.) |
Theorem | absneu 3472 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
Theorem | rabsneu 3473 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | eusn 3474* | Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.) |
Theorem | rabsnt 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.) |
Theorem | prcom 3476 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Theorem | preq1 3477 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
Theorem | preq2 3478 | Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Theorem | preq12 3479 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq1i 3480 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq2i 3481 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq12i 3482 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq1d 3483 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq2d 3484 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | preq12d 3485 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Theorem | tpeq1 3486 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Theorem | tpeq2 3487 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Theorem | tpeq3 3488 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Theorem | tpeq1d 3489 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Theorem | tpeq2d 3490 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Theorem | tpeq3d 3491 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Theorem | tpeq123d 3492 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Theorem | tprot 3493 | Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
Theorem | tpcoma 3494 | Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
Theorem | tpcomb 3495 | Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
Theorem | tpass 3496 | Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | qdass 3497 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | qdassr 3498 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | tpidm12 3499 | Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.) |
Theorem | tpidm13 3500 | Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.) |
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