Theorem List for Intuitionistic Logic Explorer - 3901-4000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | iinss 3901* |
Subset implication for an indexed intersection. (Contributed by NM,
15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
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Theorem | iinss2 3902 |
An indexed intersection is included in any of its members. (Contributed
by FL, 15-Oct-2012.)
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Theorem | uniiun 3903* |
Class union in terms of indexed union. Definition in [Stoll] p. 43.
(Contributed by NM, 28-Jun-1998.)
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Theorem | intiin 3904* |
Class intersection in terms of indexed intersection. Definition in
[Stoll] p. 44. (Contributed by NM,
28-Jun-1998.)
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Theorem | iunid 3905* |
An indexed union of singletons recovers the index set. (Contributed by
NM, 6-Sep-2005.)
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Theorem | iun0 3906 |
An indexed union of the empty set is empty. (Contributed by NM,
26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
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Theorem | 0iun 3907 |
An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
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Theorem | 0iin 3908 |
An empty indexed intersection is the universal class. (Contributed by
NM, 20-Oct-2005.)
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Theorem | viin 3909* |
Indexed intersection with a universal index class. (Contributed by NM,
11-Sep-2008.)
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Theorem | iunn0m 3910* |
There is an inhabited class in an indexed collection iff the
indexed union of them is inhabited. (Contributed by Jim Kingdon,
16-Aug-2018.)
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Theorem | iinab 3911* |
Indexed intersection of a class builder. (Contributed by NM,
6-Dec-2011.)
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Theorem | iinrabm 3912* |
Indexed intersection of a restricted class builder. (Contributed by Jim
Kingdon, 16-Aug-2018.)
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Theorem | iunin2 3913* |
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3903 to recover
Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
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Theorem | iunin1 3914* |
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3903 to recover
Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
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Theorem | iundif2ss 3915* |
Indexed union of class difference. Compare to theorem "De Morgan's
laws" in [Enderton] p. 31.
(Contributed by Jim Kingdon,
17-Aug-2018.)
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Theorem | 2iunin 3916* |
Rearrange indexed unions over intersection. (Contributed by NM,
18-Dec-2008.)
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Theorem | iindif2m 3917* |
Indexed intersection of class difference. Compare to Theorem "De
Morgan's laws" in [Enderton] p.
31. (Contributed by Jim Kingdon,
17-Aug-2018.)
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Theorem | iinin2m 3918* |
Indexed intersection of intersection. Compare to Theorem "Distributive
laws" in [Enderton] p. 30.
(Contributed by Jim Kingdon,
17-Aug-2018.)
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Theorem | iinin1m 3919* |
Indexed intersection of intersection. Compare to Theorem "Distributive
laws" in [Enderton] p. 30.
(Contributed by Jim Kingdon,
17-Aug-2018.)
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Theorem | elriin 3920* |
Elementhood in a relative intersection. (Contributed by Mario Carneiro,
30-Dec-2016.)
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Theorem | riin0 3921* |
Relative intersection of an empty family. (Contributed by Stefan
O'Rear, 3-Apr-2015.)
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Theorem | riinm 3922* |
Relative intersection of an inhabited family. (Contributed by Jim
Kingdon, 19-Aug-2018.)
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Theorem | iinxsng 3923* |
A singleton index picks out an instance of an indexed intersection's
argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario
Carneiro, 17-Nov-2016.)
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Theorem | iinxprg 3924* |
Indexed intersection with an unordered pair index. (Contributed by NM,
25-Jan-2012.)
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Theorem | iunxsng 3925* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by Mario Carneiro, 25-Jun-2016.)
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Theorem | iunxsn 3926* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro,
25-Jun-2016.)
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Theorem | iunxsngf 3927* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry
Arnoux, 2-May-2020.)
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Theorem | iunun 3928 |
Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.)
(Proof shortened by Mario Carneiro, 17-Nov-2016.)
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Theorem | iunxun 3929 |
Separate a union in the index of an indexed union. (Contributed by NM,
26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
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Theorem | iunxprg 3930* |
A pair index picks out two instances of an indexed union's argument.
(Contributed by Alexander van der Vekens, 2-Feb-2018.)
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Theorem | iunxiun 3931* |
Separate an indexed union in the index of an indexed union.
(Contributed by Mario Carneiro, 5-Dec-2016.)
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Theorem | iinuniss 3932* |
A relationship involving union and indexed intersection. Exercise 23 of
[Enderton] p. 33 but with equality
changed to subset. (Contributed by
Jim Kingdon, 19-Aug-2018.)
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Theorem | iununir 3933* |
A relationship involving union and indexed union. Exercise 25 of
[Enderton] p. 33 but with biconditional
changed to implication.
(Contributed by Jim Kingdon, 19-Aug-2018.)
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Theorem | sspwuni 3934 |
Subclass relationship for power class and union. (Contributed by NM,
18-Jul-2006.)
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Theorem | pwssb 3935* |
Two ways to express a collection of subclasses. (Contributed by NM,
19-Jul-2006.)
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Theorem | elpwpw 3936 |
Characterization of the elements of a double power class: they are exactly
the sets whose union is included in that class. (Contributed by BJ,
29-Apr-2021.)
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Theorem | pwpwab 3937* |
The double power class written as a class abstraction: the class of sets
whose union is included in the given class. (Contributed by BJ,
29-Apr-2021.)
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Theorem | pwpwssunieq 3938* |
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 3939 |
Relationship for power class and union. (Contributed by NM,
18-Jul-2006.)
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Theorem | iinpw 3940* |
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 3941* |
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 3942* |
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 3943 |
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 3944* |
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 3945* |
Alternate definition for disjoint classes. (Contributed by NM,
17-Jun-2017.)
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Disj
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Theorem | disjss2 3946 |
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 3947 |
Equality theorem for disjoint collection. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj
Disj
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Theorem | disjeq2dv 3948* |
Equality deduction for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj Disj |
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Theorem | disjss1 3949* |
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 3950* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj
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Theorem | disjeq1d 3951* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj Disj |
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Theorem | disjeq12d 3952* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj
Disj |
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Theorem | cbvdisj 3953* |
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 3954* |
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 3955* |
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 3956 |
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 3957* |
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 3958* |
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 3959* |
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 3960* |
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 3961* |
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 3962 |
Any collection of singletons is disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
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Disj |
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Theorem | 0disj 3963 |
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 3964* |
A singleton collection is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
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Disj |
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Theorem | disjx0 3965 |
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 3966 |
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 3967 |
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 4855). (Contributed by NM, 31-Dec-1993.)
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Theorem | breq 3968 |
Equality theorem for binary relations. (Contributed by NM,
4-Jun-1995.)
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Theorem | breq1 3969 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
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Theorem | breq2 3970 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
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Theorem | breq12 3971 |
Equality theorem for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqi 3972 |
Equality inference for binary relations. (Contributed by NM,
19-Feb-2005.)
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Theorem | breq1i 3973 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq2i 3974 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq12i 3975 |
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 3976 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqd 3977 |
Equality deduction for a binary relation. (Contributed by NM,
29-Oct-2011.)
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Theorem | breq2d 3978 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breq12d 3979 |
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 3980 |
Equality deduction for a binary relation. (Contributed by NM,
29-Oct-2011.)
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Theorem | breqdi 3981 |
Equality deduction for a binary relation. (Contributed by Thierry
Arnoux, 5-Oct-2020.)
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Theorem | breqan12d 3982 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | breqan12rd 3983 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
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Theorem | nbrne1 3984 |
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 3985 |
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 3986 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | eqbrtrd 3987 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 8-Oct-1999.)
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Theorem | eqbrtrri 3988 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | eqbrtrrd 3989 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | breqtri 3990 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | breqtrd 3991 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | breqtrri 3992 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 5-Aug-1993.)
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Theorem | breqtrrd 3993 |
Substitution of equal classes into a binary relation. (Contributed by
NM, 24-Oct-1999.)
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Theorem | 3brtr3i 3994 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 11-Aug-1999.)
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Theorem | 3brtr4i 3995 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 11-Aug-1999.)
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Theorem | 3brtr3d 3996 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 18-Oct-1999.)
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Theorem | 3brtr4d 3997 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 21-Feb-2005.)
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Theorem | 3brtr3g 3998 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 16-Jan-1997.)
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Theorem | 3brtr4g 3999 |
Substitution of equality into both sides of a binary relation.
(Contributed by NM, 16-Jan-1997.)
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Theorem | eqbrtrid 4000 |
B chained equality inference for a binary relation. (Contributed by NM,
11-Oct-1999.)
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