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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.)

Theoremnfopd 4605 Deduction version of bound-variable hypothesis builder nfop 4604. (Contributed by SF, 2-Jan-2015.)

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.)

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.)

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.)

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.)

Theoremcbvopab 4630* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

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.)

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

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.)

Theoremnfbr 4683 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

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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