Theorem List for Intuitionistic Logic Explorer - 3801-3900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | oteq3 3801 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
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Theorem | oteq1d 3802 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
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Theorem | oteq2d 3803 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
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Theorem | oteq3d 3804 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
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Theorem | oteq123d 3805 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
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Theorem | nfop 3806 |
Bound-variable hypothesis builder for ordered pairs. (Contributed by
NM, 14-Nov-1995.)
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Theorem | nfopd 3807 |
Deduction version of bound-variable hypothesis builder nfop 3806.
This
shows how the deduction version of a not-free theorem such as nfop 3806
can
be created from the corresponding not-free inference theorem.
(Contributed by NM, 4-Feb-2008.)
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Theorem | opid 3808 |
The ordered pair    in Kuratowski's representation.
(Contributed by FL, 28-Dec-2011.)
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Theorem | ralunsn 3809* |
Restricted quantification over the union of a set and a singleton, using
implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
(Revised by Mario Carneiro, 23-Apr-2015.)
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Theorem | 2ralunsn 3810* |
Double restricted quantification over the union of a set and a
singleton, using implicit substitution. (Contributed by Paul Chapman,
17-Nov-2012.)
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Theorem | opprc 3811 |
Expansion of an ordered pair when either member is a proper class.
(Contributed by Mario Carneiro, 26-Apr-2015.)
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Theorem | opprc1 3812 |
Expansion of an ordered pair when the first member is a proper class. See
also opprc 3811. (Contributed by NM, 10-Apr-2004.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
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Theorem | opprc2 3813 |
Expansion of an ordered pair when the second member is a proper class.
See also opprc 3811. (Contributed by NM, 15-Nov-1994.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
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Theorem | oprcl 3814 |
If an ordered pair has an element, then its arguments are sets.
(Contributed by Mario Carneiro, 26-Apr-2015.)
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Theorem | pwsnss 3815 |
The power set of a singleton. (Contributed by Jim Kingdon,
12-Aug-2018.)
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Theorem | pwpw0ss 3816 |
Compute the power set of the power set of the empty set. (See pw0 3751
for
the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with
subset in place of equality). (Contributed by Jim Kingdon,
12-Aug-2018.)
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Theorem | pwprss 3817 |
The power set of an unordered pair. (Contributed by Jim Kingdon,
13-Aug-2018.)
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Theorem | pwtpss 3818 |
The power set of an unordered triple. (Contributed by Jim Kingdon,
13-Aug-2018.)
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Theorem | pwpwpw0ss 3819 |
Compute the power set of the power set of the power set of the empty set.
(See also pw0 3751 and pwpw0ss 3816.) (Contributed by Jim Kingdon,
13-Aug-2018.)
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Theorem | pwv 3820 |
The power class of the universe is the universe. Exercise 4.12(d) of
[Mendelson] p. 235. (Contributed by NM,
14-Sep-2003.)
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2.1.18 The union of a class
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Syntax | cuni 3821 |
Extend class notation to include the union of a class. Read: "union (of)
".
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Definition | df-uni 3822* |
Define the union of a class i.e. the collection of all members of the
members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For
example,               . This is
similar to the union of two classes df-un 3145. (Contributed by NM,
23-Aug-1993.)
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Theorem | dfuni2 3823* |
Alternate definition of class union. (Contributed by NM,
28-Jun-1998.)
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Theorem | eluni 3824* |
Membership in class union. (Contributed by NM, 22-May-1994.)
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Theorem | eluni2 3825* |
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31-Aug-1999.)
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Theorem | elunii 3826 |
Membership in class union. (Contributed by NM, 24-Mar-1995.)
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Theorem | nfuni 3827 |
Bound-variable hypothesis builder for union. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | nfunid 3828 |
Deduction version of nfuni 3827. (Contributed by NM, 18-Feb-2013.)
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Theorem | csbunig 3829 |
Distribute proper substitution through the union of a class.
(Contributed by Alan Sare, 10-Nov-2012.)
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   ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)   |
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Theorem | unieq 3830 |
Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
(Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
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Theorem | unieqi 3831 |
Inference of equality of two class unions. (Contributed by NM,
30-Aug-1993.)
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Theorem | unieqd 3832 |
Deduction of equality of two class unions. (Contributed by NM,
21-Apr-1995.)
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Theorem | eluniab 3833* |
Membership in union of a class abstraction. (Contributed by NM,
11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
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Theorem | elunirab 3834* |
Membership in union of a class abstraction. (Contributed by NM,
4-Oct-2006.)
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Theorem | unipr 3835 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 23-Aug-1993.)
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Theorem | uniprg 3836 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 25-Aug-2006.)
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Theorem | unisn 3837 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 30-Aug-1993.)
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Theorem | unisng 3838 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 13-Aug-2002.)
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Theorem | dfnfc2 3839* |
An alternate statement of the effective freeness of a class , when
it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
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Theorem | uniun 3840 |
The class union of the union of two classes. Theorem 8.3 of [Quine]
p. 53. (Contributed by NM, 20-Aug-1993.)
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Theorem | uniin 3841 |
The class union of the intersection of two classes. Exercise 4.12(n) of
[Mendelson] p. 235. (Contributed by
NM, 4-Dec-2003.) (Proof shortened
by Andrew Salmon, 29-Jun-2011.)
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Theorem | uniss 3842 |
Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
(Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
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Theorem | ssuni 3843 |
Subclass relationship for class union. (Contributed by NM,
24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
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Theorem | unissi 3844 |
Subclass relationship for subclass union. Inference form of uniss 3842.
(Contributed by David Moews, 1-May-2017.)
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Theorem | unissd 3845 |
Subclass relationship for subclass union. Deduction form of uniss 3842.
(Contributed by David Moews, 1-May-2017.)
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Theorem | uni0b 3846 |
The union of a set is empty iff the set is included in the singleton of
the empty set. (Contributed by NM, 12-Sep-2004.)
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Theorem | uni0c 3847* |
The union of a set is empty iff all of its members are empty.
(Contributed by NM, 16-Aug-2006.)
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Theorem | uni0 3848 |
The union of the empty set is the empty set. Theorem 8.7 of [Quine]
p. 54. (Reproved without relying on ax-nul by Eric Schmidt.)
(Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt,
4-Apr-2007.)
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Theorem | elssuni 3849 |
An element of a class is a subclass of its union. Theorem 8.6 of [Quine]
p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40.
(Contributed by NM, 6-Jun-1994.)
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Theorem | unissel 3850 |
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18-Jul-2006.)
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Theorem | unissb 3851* |
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20-Sep-2003.)
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Theorem | uniss2 3852* |
A subclass condition on the members of two classes that implies a
subclass relation on their unions. Proposition 8.6 of [TakeutiZaring]
p. 59. (Contributed by NM, 22-Mar-2004.)
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Theorem | unidif 3853* |
If the difference
contains the largest
members of , then
the union of the difference is the union of . (Contributed by NM,
22-Mar-2004.)
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Theorem | ssunieq 3854* |
Relationship implying union. (Contributed by NM, 10-Nov-1999.)
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Theorem | unimax 3855* |
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13-Aug-2002.)
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2.1.19 The intersection of a class
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Syntax | cint 3856 |
Extend class notation to include the intersection of a class. Read:
"intersection (of) ".
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Definition | df-int 3857* |
Define the intersection of a class. Definition 7.35 of [TakeutiZaring]
p. 44. For example,             .
Compare this with the intersection of two classes, df-in 3147.
(Contributed by NM, 18-Aug-1993.)
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Theorem | dfint2 3858* |
Alternate definition of class intersection. (Contributed by NM,
28-Jun-1998.)
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Theorem | inteq 3859 |
Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
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Theorem | inteqi 3860 |
Equality inference for class intersection. (Contributed by NM,
2-Sep-2003.)
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Theorem | inteqd 3861 |
Equality deduction for class intersection. (Contributed by NM,
2-Sep-2003.)
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Theorem | elint 3862* |
Membership in class intersection. (Contributed by NM, 21-May-1994.)
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Theorem | elint2 3863* |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
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Theorem | elintg 3864* |
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20-Nov-2003.)
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Theorem | elinti 3865 |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 9-Jul-2011.)
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Theorem | nfint 3866 |
Bound-variable hypothesis builder for intersection. (Contributed by NM,
2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
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Theorem | elintab 3867* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 30-Aug-1993.)
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Theorem | elintrab 3868* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Oct-1999.)
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Theorem | elintrabg 3869* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Feb-2007.)
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Theorem | int0 3870 |
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18-Aug-1993.)
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Theorem | intss1 3871 |
An element of a class includes the intersection of the class. Exercise
4 of [TakeutiZaring] p. 44 (with
correction), generalized to classes.
(Contributed by NM, 18-Nov-1995.)
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Theorem | ssint 3872* |
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14-Oct-1999.)
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Theorem | ssintab 3873* |
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
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Theorem | ssintub 3874* |
Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
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Theorem | ssmin 3875* |
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10-Aug-2006.)
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Theorem | intmin 3876* |
Any member of a class is the smallest of those members that include it.
(Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Theorem | intss 3877 |
Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
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Theorem | intssunim 3878* |
The intersection of an inhabited set is a subclass of its union.
(Contributed by NM, 29-Jul-2006.)
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Theorem | ssintrab 3879* |
Subclass of the intersection of a restricted class builder.
(Contributed by NM, 30-Jan-2015.)
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Theorem | intssuni2m 3880* |
Subclass relationship for intersection and union. (Contributed by Jim
Kingdon, 14-Aug-2018.)
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Theorem | intminss 3881* |
Under subset ordering, the intersection of a restricted class
abstraction is less than or equal to any of its members. (Contributed
by NM, 7-Sep-2013.)
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Theorem | intmin2 3882* |
Any set is the smallest of all sets that include it. (Contributed by
NM, 20-Sep-2003.)
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Theorem | intmin3 3883* |
Under subset ordering, the intersection of a class abstraction is less
than or equal to any of its members. (Contributed by NM,
3-Jul-2005.)
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Theorem | intmin4 3884* |
Elimination of a conjunct in a class intersection. (Contributed by NM,
31-Jul-2006.)
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Theorem | intab 3885* |
The intersection of a special case of a class abstraction. may be
free in
and , which can be
thought of a    and
   . (Contributed by NM, 28-Jul-2006.) (Proof shortened by
Mario Carneiro, 14-Nov-2016.)
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Theorem | int0el 3886 |
The intersection of a class containing the empty set is empty.
(Contributed by NM, 24-Apr-2004.)
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Theorem | intun 3887 |
The class intersection of the union of two classes. Theorem 78 of
[Suppes] p. 42. (Contributed by NM,
22-Sep-2002.)
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Theorem | intpr 3888 |
The intersection of a pair is the intersection of its members. Theorem
71 of [Suppes] p. 42. (Contributed by
NM, 14-Oct-1999.)
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Theorem | intprg 3889 |
The intersection of a pair is the intersection of its members. Closed
form of intpr 3888. Theorem 71 of [Suppes] p. 42. (Contributed by FL,
27-Apr-2008.)
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Theorem | intsng 3890 |
Intersection of a singleton. (Contributed by Stefan O'Rear,
22-Feb-2015.)
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Theorem | intsn 3891 |
The intersection of a singleton is its member. Theorem 70 of [Suppes]
p. 41. (Contributed by NM, 29-Sep-2002.)
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Theorem | uniintsnr 3892* |
The union and intersection of a singleton are equal. See also eusn 3678.
(Contributed by Jim Kingdon, 14-Aug-2018.)
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Theorem | uniintabim 3893 |
The union and the intersection of a class abstraction are equal if there
is a unique satisfying value of    . (Contributed by Jim
Kingdon, 14-Aug-2018.)
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Theorem | intunsn 3894 |
Theorem joining a singleton to an intersection. (Contributed by NM,
29-Sep-2002.)
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Theorem | rint0 3895 |
Relative intersection of an empty set. (Contributed by Stefan O'Rear,
3-Apr-2015.)
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Theorem | elrint 3896* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
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Theorem | elrint2 3897* |
Membership in a restricted intersection. (Contributed by Stefan O'Rear,
3-Apr-2015.)
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2.1.20 Indexed union and
intersection
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Syntax | ciun 3898 |
Extend class notation to include indexed union. Note: Historically
(prior to 21-Oct-2005), set.mm used the notation 
 , with
the same union symbol as cuni 3821. While that syntax was unambiguous, it
did not allow for LALR parsing of the syntax constructions in set.mm. The
new syntax uses as distinguished symbol instead of and does
allow LALR parsing. Thanks to Peter Backes for suggesting this change.
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Syntax | ciin 3899 |
Extend class notation to include indexed intersection. Note:
Historically (prior to 21-Oct-2005), set.mm used the notation
  , with the
same intersection symbol as cint 3856. Although
that syntax was unambiguous, it did not allow for LALR parsing of the
syntax constructions in set.mm. The new syntax uses a distinguished
symbol
instead of and
does allow LALR parsing. Thanks to
Peter Backes for suggesting this change.
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Definition | df-iun 3900* |
Define indexed union. Definition indexed union in [Stoll] p. 45. In
most applications, is independent of (although this is not
required by the definition), and depends on i.e. can be read
informally as    . We call the index, the index
set, and the
indexed set. In most books, is written as
a subscript or underneath a union symbol . We use a special
union symbol to make it easier to distinguish from plain class
union. In many theorems, you will see that and are in the
same disjoint variable group (meaning cannot depend on ) and
that and do not share a disjoint
variable group (meaning
that can be thought of as    i.e. can be substituted with a
class expression containing ). An alternate definition tying
indexed union to ordinary union is dfiun2 3932. Theorem uniiun 3952 provides
a definition of ordinary union in terms of indexed union. (Contributed
by NM, 27-Jun-1998.)
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