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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

2.1.21  Disjointness

Syntaxwdisj 3901 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj

Definitiondf-disj 3902* A collection of classes is disjoint when for each element , it is in for at most one . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
Disj

Theoremdfdisj2 3903* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Disj

Theoremdisjss2 3904 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2 3905 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2dv 3906* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjss1 3907* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1 3908* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1d 3909* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq12d 3910* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisj 3911* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisjv 3912* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Disj Disj

Theoremnfdisjv 3913* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
Disj

Theoremnfdisj1 3914 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjnim 3915* If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
Disj

Theoremdisjnims 3916* If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
Disj

Theoremdisji2 3917* Property of a disjoint collection: if and , and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoreminvdisj 3918* If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremdisjiun 3919* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremsndisj 3920 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theorem0disj 3921 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxsn 3922* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjx0 3923 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

2.1.22  Binary relations

Syntaxwbr 3924 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 3925 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. This definition of relations 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 (see for example iprc 4802). (Contributed by NM, 31-Dec-1993.)

Theorembreq 3926 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Theorembreq1 3927 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq2 3928 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq12 3929 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqi 3930 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)

Theorembreq1i 3931 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq2i 3932 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12i 3933 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theorembreq1d 3934 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqd 3935 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreq2d 3936 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12d 3937 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorembreq123d 3938 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreqdi 3939 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)

Theorembreqan12d 3940 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqan12rd 3941 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theoremnbrne1 3942 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremnbrne2 3943 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremeqbrtri 3944 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrd 3945 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)

Theoremeqbrtrri 3946 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrrd 3947 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtri 3948 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrd 3949 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtrri 3950 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrrd 3951 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorem3brtr3i 3952 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr4i 3953 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr3d 3954 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)

Theorem3brtr4d 3955 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)

Theorem3brtr3g 3956 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theorem3brtr4g 3957 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Theoremeqbrtrid 3958 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theoremeqbrtrrid 3959 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Theorembreqtrid 3960 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theorembreqtrrid 3961 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremeqbrtrdi 3962 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Theoremeqbrtrrdi 3963 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Theorembreqtrdi 3964 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Theorembreqtrrdi 3965 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Theoremssbrd 3966 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Theoremssbri 3967 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremnfbrd 3968 Deduction version of bound-variable hypothesis builder nfbr 3969. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremnfbr 3969 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theorembrab1 3970* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Theorembr0 3971 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)

Theorembrne0 3972 If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 3973. (Contributed by Alexander van der Vekens, 7-Jul-2018.)

Theorembrm 3973* If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)

Theorembrun 3974 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Theorembrin 3975 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Theorembrdif 3976 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremsbcbrg 3977 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbcbr12g 3978* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr1g 3979* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theoremsbcbr2g 3980* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)

Theorembrralrspcev 3981* Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)

Theorembrimralrspcev 3982* Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)

2.1.23  Ordered-pair class abstractions (class builders)

Syntaxcopab 3983 Extend class notation to include ordered-pair class abstraction (class builder).

Syntaxcmpt 3984 Extend the definition of a class to include maps-to notation for defining a function via a rule.

Definitiondf-opab 3985* 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. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)

Definitiondf-mpt 3986* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from (in ) to ." The class expression is the value of the function at and normally contains the variable . Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)

Theoremopabss 3987* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbid 3988 Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremopabbidv 3989* Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)

Theoremopabbii 3990 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Theoremnfopab 3991* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremnfopab1 3992 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 3993 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 3994* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Theoremcbvopabv 3995* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Theoremcbvopab1 3996* 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 3997* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvopab1s 3998* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvopab1v 3999* 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 4000* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

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