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Theorem List for New Foundations Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremproj2op 4601 The second projection operator applied to an ordered pair yields its second member. Theorem X.2.8 of [Rosser] p. 283. (Contributed by SF, 3-Feb-2015.)
Proj2
 
Theoremopth 4602 The ordered pair theorem. Two ordered pairs are equal iff their components are equal. (Contributed by SF, 2-Jan-2015.)
 
Theoremopexb 4603 An ordered pair is a set iff its components are sets. (Contributed by SF, 2-Jan-2015.)
 
Theoremnfop 4604 Bound-variable hypothesis builder for ordered pairs. (Contributed by SF, 2-Jan-2015.)
 F/_   &     F/_   =>     F/_
 
Theoremnfopd 4605 Deduction version of bound-variable hypothesis builder nfop 4604. (Contributed by SF, 2-Jan-2015.)
 F/_   &     F/_   =>     F/_
 
Theoremeqvinop 4606* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
   &       =>   
 
Theoremcopsexg 4607* Substitution of class for ordered pair . (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 25-Jul-2011.)
 
Theoremcopsex2t 4608* Closed theorem form of copsex2g 4609. (Contributed by NM, 17-Feb-2013.)
 
Theoremcopsex2g 4609* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
   =>   
 
Theoremcopsex4g 4610* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
   =>   
 
Theoremeqop 4611* Express equality to an ordered pair. (Contributed by SF, 6-Jan-2015.)
Phi Phi 0c
 
Theoremmosubopt 4612* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
 
Theoremmosubop 4613* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
   =>   
 
Theoremphiun 4614 The phi operation distributes over union. (Contributed by SF, 20-Feb-2015.)
Phi Phi Phi
 
Theoremphidisjnn 4615 The phi operation applied to a set disjoint from the naturals has no effect. (Contributed by SF, 20-Feb-2015.)
Nn Phi
 
Theoremphialllem1 4616* Lemma for phiall 4618. Any set of numbers without zero is the Phi of a set. (Contributed by Scott Fenton, 14-Apr-2021.)
   =>    Nn 0c Phi
 
Theoremphialllem2 4617* Lemma for phiall 4618. Any set without 0c is equal to the Phi of a set. (Contributed by Scott Fenton, 8-Apr-2021.)
   =>    0c Phi
 
Theoremphiall 4618* Any set is equal to either the Phi of another set or to a Phi with 0c adjoined. (Contributed by Scott Fenton, 8-Apr-2021.)
   =>    Phi Phi 0c
 
Theoremopeq 4619 Any class is equal to an ordered pair. (Contributed by Scott Fenton, 8-Apr-2021.)
Proj1 Proj2
 
Theoremopeqexb 4620* A class is a set iff it is equal to an ordered pair. (Contributed by Scott Fenton, 19-Apr-2021.)
 
Theoremopeqex 4621* Any set is equal to some ordered pair. (Contributed by Scott Fenton, 16-Apr-2021.)
 
2.3.2  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 4622 Extend class notation to include ordered-pair class abstraction (class builder).
 
Definitiondf-opab 4623* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually and are distinct, although the definition doesn't strictly require it (see dfid2 4769 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by SF, 12-Jan-2015.)
 
Theoremopabbid 4624 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 F/   &     F/   &       =>   
 
Theoremopabbidv 4625* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
   =>   
 
Theoremopabbii 4626 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
   =>   
 
Theoremnfopab 4627* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

 F/   =>     F/_
 
Theoremnfopab1 4628 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_
 
Theoremnfopab2 4629 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_
 
Theoremcbvopab 4630* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

 F/   &     F/   &     F/   &     F/   &       =>   
 
Theoremcbvopabv 4631* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
   =>   
 
Theoremcbvopab1 4632* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

 F/   &     F/   &       =>   
 
Theoremcbvopab2 4633* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

 F/   &     F/   &       =>   
 
Theoremcbvopab1s 4634* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
 
Theoremcbvopab1v 4635* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
   =>   
 
Theoremcbvopab2v 4636* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
   =>   
 
Theoremcsbopabg 4637* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

 [.  ].
 
Theoremunopab 4638 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 
2.3.3  Binary relations
 
Syntaxwbr 4639 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
 
Definitiondf-br 4640 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class normally denotes a relation that compares two classes and . This definition is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when is a proper class. (Contributed by NM, 4-Jun-1995.)
 
Theorembreq 4641 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 
Theorembreq1 4642 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 
Theorembreq2 4643 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 
Theorembreq12 4644 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 
Theorembreqi 4645 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
   =>   
 
Theorembreq1i 4646 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>   
 
Theorembreq2i 4647 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>   
 
Theorembreq12i 4648 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Revised by Eric Schmidt, 4-Apr-2007.)
   &       =>   
 
Theorembreq1d 4649 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>   
 
Theorembreqd 4650 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
   =>   
 
Theorembreq2d 4651 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>   
 
Theorembreq12d 4652 Equality deduction for a binary relation. (The proof was shortened by Andrew Salmon, 9-Jul-2011.) (Contributed by NM, 8-Feb-1996.) (Revised by set.mm contributors, 9-Jul-2011.)
   &       =>   
 
Theorembreq123d 4653 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
   &       &       =>   
 
Theorembreqan12d 4654 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   &       =>   
 
Theorembreqan12rd 4655 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   &       =>   
 
Theoremnbrne1 4656 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 
Theoremnbrne2 4657 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 
Theoremeqbrtri 4658 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremeqbrtrd 4659 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
   &       =>   
 
Theoremeqbrtrri 4660 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremeqbrtrrd 4661 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
   &       =>   
 
Theorembreqtri 4662 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theorembreqtrd 4663 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
   &       =>   
 
Theorembreqtrri 4664 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theorembreqtrrd 4665 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
   &       =>   
 
Theorem3brtr3i 4666 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
   &       &       =>   
 
Theorem3brtr4i 4667 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
   &       &       =>   
 
Theorem3brtr3d 4668 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
   &       &       =>   
 
Theorem3brtr4d 4669 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
   &       &       =>   
 
Theorem3brtr3g 4670 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
   &       &       =>   
 
Theorem3brtr4g 4671 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
   &       &       =>   
 
Theoremsyl5eqbr 4672 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
   &       =>   
 
Theoremsyl5eqbrr 4673 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
   &       =>   
 
Theoremsyl5breq 4674 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
   &       =>   
 
Theoremsyl5breqr 4675 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
   &       =>   
 
Theoremsyl6eqbr 4676 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
   &       =>   
 
Theoremsyl6eqbrr 4677 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
   &       =>   
 
Theoremsyl6breq 4678 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
   &       =>   
 
Theoremsyl6breqr 4679 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
   &       =>   
 
Theoremssbrd 4680 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
   =>   
 
Theoremssbri 4681 Inference from a subclass relationship of binary relations. (The proof was shortened by Andrew Salmon, 9-Jul-2011.) (Contributed by NM, 28-Mar-2007.) (Revised by set.mm contributors, 9-Jul-2011.)
   =>   
 
Theoremnfbrd 4682 Deduction version of bound-variable hypothesis builder nfbr 4683. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   &     F/_   =>     F/
 
Theoremnfbr 4683 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   &     F/_   =>    
 F/
 
Theorembrab1 4684* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
 
Theoremsbcbrg 4685 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 [.  ].
 
Theoremsbcbr12g 4686* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 [.  ].
 
Theoremsbcbr1g 4687* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 [.  ].
 
Theoremsbcbr2g 4688* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 [.  ].
 
Theorembrex 4689 Binary relationship implies sethood of both parts. (Contributed by SF, 7-Jan-2015.)
 
Theorembrreldmex 4690 Binary relationship implies sethood of domain. (Contributed by SF, 7-Jan-2018.)
 
Theorembrrelrnex 4691 Binary relationship implies sethood of range. (Contributed by SF, 7-Jan-2018.)
 
Theorembrun 4692 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 
Theorembrin 4693 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 
Theorembrdif 4694 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4695 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (The proof was shortened by Andrew Salmon, 25-Jul-2011.) (Contributed by NM, 14-Apr-1995.) (Revised by set.mm contributors, 25-Jul-2011.)
 
Theoremelopab 4696* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
 
Theoremopelopabsb 4697* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
 [.  ]. [.  ].
 
Theorembrabsb 4698* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
   =>     [.  ]. [.  ].
 
Theoremopelopabt 4699* Closed theorem form of opelopab 4708. (Contributed by NM, 19-Feb-2013.)
 
Theoremopelopabga 4700* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
   =>   
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