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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-tp 3501 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
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Definition | df-op 3502* |
Definition of an ordered pair, equivalent to Kuratowski's definition
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Definition 9.1 of [Quine] p. 58 defines an
ordered pair unconditionally
as
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
|
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Definition | df-ot 3503 | Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.) |
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Theorem | sneq 3504 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sneqi 3505 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
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Theorem | sneqd 3506 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
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Theorem | dfsn2 3507 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
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Theorem | elsng 3508 | 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.) |
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Theorem | elsn 3509 | There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
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Theorem | velsn 3510 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
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Theorem | elsni 3511 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
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Theorem | dfpr2 3512* | Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
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Theorem | elprg 3513 | 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.) |
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Theorem | elpr 3514 | 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.) |
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Theorem | elpr2 3515 | 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.) |
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Theorem | elpri 3516 | 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.) |
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Theorem | nelpri 3517 | 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.) |
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Theorem | prneli 3518 |
If an element doesn't match the items in an unordered pair, it is not in
the unordered pair, using ![]() |
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Theorem | nelprd 3519 | 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.) |
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Theorem | snidg 3520 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.) |
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Theorem | snidb 3521 | A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
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Theorem | snid 3522 | A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.) |
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Theorem | vsnid 3523 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | elsn2g 3524 |
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. This variation requires only that ![]() ![]() |
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Theorem | elsn2 3525 |
There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. This variation requires only that ![]() ![]() |
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Theorem | mosn 3526* |
A singleton has at most one element. This works whether ![]() |
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Theorem | ralsnsg 3527* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ralsns 3528* | Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexsns 3529* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.) |
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Theorem | ralsng 3530* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexsng 3531* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
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Theorem | exsnrex 3532 | 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.) |
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Theorem | ralsn 3533* | Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.) |
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Theorem | rexsn 3534* | Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | eltpg 3535 | Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
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Theorem | eltpi 3536 | A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
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Theorem | eltp 3537 | 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.) |
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Theorem | dftp2 3538* | Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.) |
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Theorem | nfpr 3539 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
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Theorem | ralprg 3540* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexprg 3541* | Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | raltpg 3542* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rextpg 3543* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ralpr 3544* | Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rexpr 3545* | Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | raltp 3546* | Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rextp 3547* | Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | sbcsng 3548* | Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
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Theorem | nfsn 3549 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
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Theorem | csbsng 3550 | Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
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Theorem | disjsn 3551 | 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.) |
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Theorem | disjsn2 3552 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
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Theorem | disjpr2 3553 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
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Theorem | snprc 3554 | 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.) |
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Theorem | r19.12sn 3555* | Special case of r19.12 2512 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.) |
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Theorem | rabsn 3556* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
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Theorem | rabrsndc 3557* | 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.) |
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Theorem | euabsn2 3558* | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
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Theorem | euabsn 3559 | Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
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Theorem | reusn 3560* | 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.) |
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Theorem | absneu 3561 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
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Theorem | rabsneu 3562 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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Theorem | eusn 3563* |
Two ways to express "![]() |
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Theorem | rabsnt 3564* | 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.) |
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Theorem | prcom 3565 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
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Theorem | preq1 3566 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
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Theorem | preq2 3567 | Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
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Theorem | preq12 3568 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq1i 3569 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq2i 3570 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq12i 3571 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq1d 3572 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq2d 3573 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | preq12d 3574 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
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Theorem | tpeq1 3575 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq2 3576 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq3 3577 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
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Theorem | tpeq1d 3578 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq2d 3579 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq3d 3580 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tpeq123d 3581 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
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Theorem | tprot 3582 | Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
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Theorem | tpcoma 3583 | Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
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Theorem | tpcomb 3584 | Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
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Theorem | tpass 3585 | Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | qdass 3586 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | qdassr 3587 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
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Theorem | tpidm12 3588 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm13 3589 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm23 3590 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tpidm 3591 |
Unordered triple ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tppreq3 3592 | An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.) |
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Theorem | prid1g 3593 | An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
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Theorem | prid2g 3594 | An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
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Theorem | prid1 3595 | An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) |
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Theorem | prid2 3596 | An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) |
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Theorem | prprc1 3597 | A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.) |
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Theorem | prprc2 3598 | A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.) |
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Theorem | prprc 3599 | An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.) |
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Theorem | tpid1 3600 | One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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