Theorem List for Intuitionistic Logic Explorer - 3901-4000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | pwpwssunieq 3901* |
The class of sets whose union is equal to a given class is included in
the double power class of that class. (Contributed by BJ,
29-Apr-2021.)
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Theorem | elpwuni 3902 |
Relationship for power class and union. (Contributed by NM,
18-Jul-2006.)
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Theorem | iinpw 3903* |
The power class of an intersection in terms of indexed intersection.
Exercise 24(a) of [Enderton] p. 33.
(Contributed by NM,
29-Nov-2003.)
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Theorem | iunpwss 3904* |
Inclusion of an indexed union of a power class in the power class of the
union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
(Contributed by NM, 25-Nov-2003.)
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Theorem | rintm 3905* |
Relative intersection of an inhabited class. (Contributed by Jim
Kingdon, 19-Aug-2018.)
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2.1.21 Disjointness
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Syntax | wdisj 3906 |
Extend wff notation to include the statement that a family of classes
, for , is a disjoint family.
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Disj |
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Definition | df-disj 3907* |
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.)
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Disj
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Theorem | dfdisj2 3908* |
Alternate definition for disjoint classes. (Contributed by NM,
17-Jun-2017.)
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Disj
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Theorem | disjss2 3909 |
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.)
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Disj
Disj |
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Theorem | disjeq2 3910 |
Equality theorem for disjoint collection. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj
Disj
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Theorem | disjeq2dv 3911* |
Equality deduction for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj Disj |
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Theorem | disjss1 3912* |
A subset of a disjoint collection is disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj |
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Theorem | disjeq1 3913* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj
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Theorem | disjeq1d 3914* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj Disj |
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Theorem | disjeq12d 3915* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj |
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Theorem | cbvdisj 3916* |
Change bound variables in a disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj |
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Theorem | cbvdisjv 3917* |
Change bound variables in a disjoint collection. (Contributed by Mario
Carneiro, 11-Dec-2016.)
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Disj Disj |
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Theorem | nfdisjv 3918* |
Bound-variable hypothesis builder for disjoint collection. (Contributed
by Jim Kingdon, 19-Aug-2018.)
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Disj |
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Theorem | nfdisj1 3919 |
Bound-variable hypothesis builder for disjoint collection. (Contributed
by Mario Carneiro, 14-Nov-2016.)
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Disj
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Theorem | disjnim 3920* |
If a collection for is disjoint, then pairs are
disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by
Jim Kingdon, 6-Oct-2022.)
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Disj
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Theorem | disjnims 3921* |
If a collection for is disjoint, then pairs are
disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by
Jim Kingdon, 7-Oct-2022.)
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Disj
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Theorem | disji2 3922* |
Property of a disjoint collection: if and
, and , then and
are disjoint.
(Contributed by Mario Carneiro, 14-Nov-2016.)
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Disj
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Theorem | invdisj 3923* |
If there is a function such that for all
, then the sets for distinct
are
disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
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Disj |
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Theorem | disjiun 3924* |
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.)
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Disj
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Theorem | sndisj 3925 |
Any collection of singletons is disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj |
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Theorem | 0disj 3926 |
Any collection of empty sets is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj |
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Theorem | disjxsn 3927* |
A singleton collection is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj |
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Theorem | disjx0 3928 |
An empty collection is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj |
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2.1.22 Binary relations
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Syntax | wbr 3929 |
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.
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Definition | df-br 3930 |
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 4807). (Contributed by NM, 31-Dec-1993.)
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Theorem | breq 3931 |
Equality theorem for binary relations. (Contributed by NM,
4-Jun-1995.)
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Theorem | breq1 3932 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
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Theorem | breq2 3933 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
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Theorem | breq12 3934 |
Equality theorem for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqi 3935 |
Equality inference for binary relations. (Contributed by NM,
19-Feb-2005.)
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Theorem | breq1i 3936 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq2i 3937 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq12i 3938 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
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Theorem | breq1d 3939 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqd 3940 |
Equality deduction for a binary relation. (Contributed by NM,
29-Oct-2011.)
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Theorem | breq2d 3941 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq12d 3942 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
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Theorem | breq123d 3943 |
Equality deduction for a binary relation. (Contributed by NM,
29-Oct-2011.)
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Theorem | breqdi 3944 |
Equality deduction for a binary relation. (Contributed by Thierry
Arnoux, 5-Oct-2020.)
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Theorem | breqan12d 3945 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqan12rd 3946 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | nbrne1 3947 |
Two classes are different if they don't have the same relationship to a
third class. (Contributed by NM, 3-Jun-2012.)
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Theorem | nbrne2 3948 |
Two classes are different if they don't have the same relationship to a
third class. (Contributed by NM, 3-Jun-2012.)
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Theorem | eqbrtri 3949 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | eqbrtrd 3950 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 8-Oct-1999.)
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Theorem | eqbrtrri 3951 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | eqbrtrrd 3952 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | breqtri 3953 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | breqtrd 3954 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | breqtrri 3955 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | breqtrrd 3956 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | 3brtr3i 3957 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 11-Aug-1999.)
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Theorem | 3brtr4i 3958 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 11-Aug-1999.)
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Theorem | 3brtr3d 3959 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 18-Oct-1999.)
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Theorem | 3brtr4d 3960 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 21-Feb-2005.)
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Theorem | 3brtr3g 3961 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 16-Jan-1997.)
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Theorem | 3brtr4g 3962 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 16-Jan-1997.)
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Theorem | eqbrtrid 3963 |
B chained equality inference for a binary relation. (Contributed by NM,
11-Oct-1999.)
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Theorem | eqbrtrrid 3964 |
B chained equality inference for a binary relation. (Contributed by NM,
17-Sep-2004.)
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Theorem | breqtrid 3965 |
B chained equality inference for a binary relation. (Contributed by NM,
11-Oct-1999.)
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Theorem | breqtrrid 3966 |
B chained equality inference for a binary relation. (Contributed by NM,
24-Apr-2005.)
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Theorem | eqbrtrdi 3967 |
A chained equality inference for a binary relation. (Contributed by NM,
12-Oct-1999.)
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Theorem | eqbrtrrdi 3968 |
A chained equality inference for a binary relation. (Contributed by NM,
4-Jan-2006.)
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Theorem | breqtrdi 3969 |
A chained equality inference for a binary relation. (Contributed by NM,
11-Oct-1999.)
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Theorem | breqtrrdi 3970 |
A chained equality inference for a binary relation. (Contributed by NM,
24-Apr-2005.)
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Theorem | ssbrd 3971 |
Deduction from a subclass relationship of binary relations.
(Contributed by NM, 30-Apr-2004.)
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Theorem | ssbri 3972 |
Inference from a subclass relationship of binary relations.
(Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro,
8-Feb-2015.)
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Theorem | nfbrd 3973 |
Deduction version of bound-variable hypothesis builder nfbr 3974.
(Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro,
14-Oct-2016.)
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Theorem | nfbr 3974 |
Bound-variable hypothesis builder for binary relation. (Contributed by
NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
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Theorem | brab1 3975* |
Relationship between a binary relation and a class abstraction.
(Contributed by Andrew Salmon, 8-Jul-2011.)
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Theorem | br0 3976 |
The empty binary relation never holds. (Contributed by NM,
23-Aug-2018.)
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Theorem | brne0 3977 |
If two sets are in a binary relation, the relation cannot be empty. In
fact, the relation is also inhabited, as seen at brm 3978.
(Contributed by
Alexander van der Vekens, 7-Jul-2018.)
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Theorem | brm 3978* |
If two sets are in a binary relation, the relation is inhabited.
(Contributed by Jim Kingdon, 31-Dec-2023.)
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Theorem | brun 3979 |
The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
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Theorem | brin 3980 |
The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
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Theorem | brdif 3981 |
The difference of two binary relations. (Contributed by Scott Fenton,
11-Apr-2011.)
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Theorem | sbcbrg 3982 |
Move substitution in and out of a binary relation. (Contributed by NM,
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
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Theorem | sbcbr12g 3983* |
Move substitution in and out of a binary relation. (Contributed by NM,
13-Dec-2005.)
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Theorem | sbcbr1g 3984* |
Move substitution in and out of a binary relation. (Contributed by NM,
13-Dec-2005.)
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Theorem | sbcbr2g 3985* |
Move substitution in and out of a binary relation. (Contributed by NM,
13-Dec-2005.)
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Theorem | brralrspcev 3986* |
Restricted existential specialization with a restricted universal
quantifier over a relation, closed form. (Contributed by AV,
20-Aug-2022.)
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Theorem | brimralrspcev 3987* |
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.)
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2.1.23 Ordered-pair class abstractions (class
builders)
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Syntax | copab 3988 |
Extend class notation to include ordered-pair class abstraction (class
builder).
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Syntax | cmpt 3989 |
Extend the definition of a class to include maps-to notation for defining
a function via a rule.
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Definition | df-opab 3990* |
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.)
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Definition | df-mpt 3991* |
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.)
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Theorem | opabss 3992* |
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.)
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Theorem | opabbid 3993 |
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.)
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Theorem | opabbidv 3994* |
Equivalent wff's yield equal ordered-pair class abstractions (deduction
form). (Contributed by NM, 15-May-1995.)
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Theorem | opabbii 3995 |
Equivalent wff's yield equal class abstractions. (Contributed by NM,
15-May-1995.)
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Theorem | nfopab 3996* |
Bound-variable hypothesis builder for class abstraction. (Contributed
by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by
Andrew Salmon, 11-Jul-2011.)
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Theorem | nfopab1 3997 |
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.)
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Theorem | nfopab2 3998 |
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.)
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Theorem | cbvopab 3999* |
Rule used to change bound variables in an ordered-pair class
abstraction, using implicit substitution. (Contributed by NM,
14-Sep-2003.)
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Theorem | cbvopabv 4000* |
Rule used to change bound variables in an ordered-pair class
abstraction, using implicit substitution. (Contributed by NM,
15-Oct-1996.)
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