Theorem List for Intuitionistic Logic Explorer - 4401-4500 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | reliun 4401 |
An indexed union is a relation iff each member of its indexed family is
a relation. (Contributed by NM, 19-Dec-2008.)
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Theorem | reliin 4402 |
An indexed intersection is a relation if at least one of the member of the
indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
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Theorem | reluni 4403* |
The union of a class is a relation iff any member is a relation.
Exercise 6 of [TakeutiZaring] p.
25 and its converse. (Contributed by
NM, 13-Aug-2004.)
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Theorem | relint 4404* |
The intersection of a class is a relation if at least one member is a
relation. (Contributed by NM, 8-Mar-2014.)
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Theorem | rel0 4405 |
The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
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Theorem | relopabi 4406 |
A class of ordered pairs is a relation. (Contributed by Mario Carneiro,
21-Dec-2013.)
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Theorem | relopab 4407 |
A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.)
(Unnecessary distinct variable restrictions were removed by Alan Sare,
9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
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Theorem | reli 4408 |
The identity relation is a relation. Part of Exercise 4.12(p) of
[Mendelson] p. 235. (Contributed by
NM, 26-Apr-1998.) (Revised by
Mario Carneiro, 21-Dec-2013.)
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Theorem | rele 4409 |
The membership relation is a relation. (Contributed by NM,
26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | opabid2 4410* |
A relation expressed as an ordered pair abstraction. (Contributed by
NM, 11-Dec-2006.)
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Theorem | inopab 4411* |
Intersection of two ordered pair class abstractions. (Contributed by
NM, 30-Sep-2002.)
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Theorem | difopab 4412* |
The difference of two ordered-pair abstractions. (Contributed by Stefan
O'Rear, 17-Jan-2015.)
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Theorem | inxp 4413 |
The intersection of two cross products. Exercise 9 of [TakeutiZaring]
p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | xpindi 4414 |
Distributive law for cross product over intersection. Theorem 102 of
[Suppes] p. 52. (Contributed by NM,
26-Sep-2004.)
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Theorem | xpindir 4415 |
Distributive law for cross product over intersection. Similar to
Theorem 102 of [Suppes] p. 52.
(Contributed by NM, 26-Sep-2004.)
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Theorem | xpiindim 4416* |
Distributive law for cross product over indexed intersection.
(Contributed by Jim Kingdon, 7-Dec-2018.)
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Theorem | xpriindim 4417* |
Distributive law for cross product over relativized indexed
intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
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Theorem | eliunxp 4418* |
Membership in a union of cross products. Analogue of elxp 4305
for
nonconstant    . (Contributed by Mario Carneiro,
29-Dec-2014.)
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Theorem | opeliunxp2 4419* |
Membership in a union of cross products. (Contributed by Mario
Carneiro, 14-Feb-2015.)
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Theorem | raliunxp 4420* |
Write a double restricted quantification as one universal quantifier.
In this version of ralxp 4422,    is not assumed to be
constant.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | rexiunxp 4421* |
Write a double restricted quantification as one universal quantifier.
In this version of rexxp 4423,    is not assumed to be
constant.
(Contributed by Mario Carneiro, 14-Feb-2015.)
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Theorem | ralxp 4422* |
Universal quantification restricted to a cross product is equivalent to
a double restricted quantification. The hypothesis specifies an
implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by
Mario Carneiro, 29-Dec-2014.)
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Theorem | rexxp 4423* |
Existential quantification restricted to a cross product is equivalent
to a double restricted quantification. (Contributed by NM,
11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
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Theorem | djussxp 4424* |
Disjoint union is a subset of a cross product. (Contributed by Stefan
O'Rear, 21-Nov-2014.)
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Theorem | ralxpf 4425* |
Version of ralxp 4422 with bound-variable hypotheses. (Contributed
by NM,
18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | rexxpf 4426* |
Version of rexxp 4423 with bound-variable hypotheses. (Contributed
by NM,
19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | iunxpf 4427* |
Indexed union on a cross product is equals a double indexed union. The
hypothesis specifies an implicit substitution. (Contributed by NM,
19-Dec-2008.)
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Theorem | opabbi2dv 4428* |
Deduce equality of a relation and an ordered-pair class builder.
Compare abbi2dv 2153. (Contributed by NM, 24-Feb-2014.)
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Theorem | relop 4429* |
A necessary and sufficient condition for a Kuratowski ordered pair to be
a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this
detail.)
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Theorem | ideqg 4430 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | ideq 4431 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 13-Aug-1995.)
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Theorem | ididg 4432 |
A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | issetid 4433 |
Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario
Carneiro, 26-Apr-2015.)
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Theorem | coss1 4434 |
Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
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Theorem | coss2 4435 |
Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
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Theorem | coeq1 4436 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq2 4437 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq1i 4438 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2i 4439 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq1d 4440 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2d 4441 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq12i 4442 |
Equality inference for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | coeq12d 4443 |
Equality deduction for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | nfco 4444 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 1-Sep-1999.)
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Theorem | brcog 4445* |
Ordered pair membership in a composition. (Contributed by NM,
24-Feb-2015.)
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Theorem | opelco2g 4446* |
Ordered pair membership in a composition. (Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | brcogw 4447 |
Ordered pair membership in a composition. (Contributed by Thierry
Arnoux, 14-Jan-2018.)
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Theorem | eqbrrdva 4448* |
Deduction from extensionality principle for relations, given an
equivalence only on the relation's domain and range. (Contributed by
Thierry Arnoux, 28-Nov-2017.)
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Theorem | brco 4449* |
Binary relation on a composition. (Contributed by NM, 21-Sep-2004.)
(Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | opelco 4450* |
Ordered pair membership in a composition. (Contributed by NM,
27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | cnvss 4451 |
Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
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Theorem | cnveq 4452 |
Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
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Theorem | cnveqi 4453 |
Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
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Theorem | cnveqd 4454 |
Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
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Theorem | elcnv 4455* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 24-Mar-1998.)
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Theorem | elcnv2 4456* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 11-Aug-2004.)
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Theorem | nfcnv 4457 |
Bound-variable hypothesis builder for converse. (Contributed by NM,
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | opelcnvg 4458 |
Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | brcnvg 4459 |
The converse of a binary relation swaps arguments. Theorem 11 of [Suppes]
p. 61. (Contributed by NM, 10-Oct-2005.)
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Theorem | opelcnv 4460 |
Ordered-pair membership in converse. (Contributed by NM,
13-Aug-1995.)
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Theorem | brcnv 4461 |
The converse of a binary relation swaps arguments. Theorem 11 of
[Suppes] p. 61. (Contributed by NM,
13-Aug-1995.)
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Theorem | csbcnvg 4462 |
Move class substitution in and out of the converse of a function.
(Contributed by Thierry Arnoux, 8-Feb-2017.)
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    ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    |
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Theorem | cnvco 4463 |
Distributive law of converse over class composition. Theorem 26 of
[Suppes] p. 64. (Contributed by NM,
19-Mar-1998.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | cnvuni 4464* |
The converse of a class union is the (indexed) union of the converses of
its members. (Contributed by NM, 11-Aug-2004.)
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Theorem | dfdm3 4465* |
Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfrn2 4466* |
Alternate definition of range. Definition 4 of [Suppes] p. 60.
(Contributed by NM, 27-Dec-1996.)
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Theorem | dfrn3 4467* |
Alternate definition of range. Definition 6.5(2) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | elrn2g 4468* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | elrng 4469* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | dfdm4 4470 |
Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfdmf 4471* |
Definition of domain, using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by NM, 8-Mar-1995.)
(Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | csbdmg 4472 |
Distribute proper substitution through the domain of a class.
(Contributed by Jim Kingdon, 8-Dec-2018.)
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   ![]_ ]_](_urbrack.gif)
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Theorem | eldmg 4473* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by Mario
Carneiro, 9-Jul-2014.)
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Theorem | eldm2g 4474* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
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Theorem | eldm 4475* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 2-Apr-2004.)
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Theorem | eldm2 4476* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 1-Aug-1994.)
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Theorem | dmss 4477 |
Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeq 4478 |
Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeqi 4479 |
Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | dmeqd 4480 |
Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | opeldm 4481 |
Membership of first of an ordered pair in a domain. (Contributed by NM,
30-Jul-1995.)
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Theorem | breldm 4482 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 30-Jul-1995.)
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Theorem | opeldmg 4483 |
Membership of first of an ordered pair in a domain. (Contributed by Jim
Kingdon, 9-Jul-2019.)
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Theorem | breldmg 4484 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 21-Mar-2007.)
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Theorem | dmun 4485 |
The domain of a union is the union of domains. Exercise 56(a) of
[Enderton] p. 65. (Contributed by NM,
12-Aug-1994.) (Proof shortened
by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmin 4486 |
The domain of an intersection belong to the intersection of domains.
Theorem 6 of [Suppes] p. 60.
(Contributed by NM, 15-Sep-2004.)
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Theorem | dmiun 4487 |
The domain of an indexed union. (Contributed by Mario Carneiro,
26-Apr-2016.)
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Theorem | dmuni 4488* |
The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
(Contributed by NM, 3-Feb-2004.)
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Theorem | dmopab 4489* |
The domain of a class of ordered pairs. (Contributed by NM,
16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
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Theorem | dmopabss 4490* |
Upper bound for the domain of a restricted class of ordered pairs.
(Contributed by NM, 31-Jan-2004.)
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Theorem | dmopab3 4491* |
The domain of a restricted class of ordered pairs. (Contributed by NM,
31-Jan-2004.)
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Theorem | dm0 4492 |
The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmi 4493 |
The domain of the identity relation is the universe. (Contributed by
NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmv 4494 |
The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.)
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Theorem | dm0rn0 4495 |
An empty domain implies an empty range. (Contributed by NM,
21-May-1998.)
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Theorem | reldm0 4496 |
A relation is empty iff its domain is empty. (Contributed by NM,
15-Sep-2004.)
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Theorem | dmmrnm 4497* |
A domain is inhabited if and only if the range is inhabited.
(Contributed by Jim Kingdon, 15-Dec-2018.)
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Theorem | dmxpm 4498* |
The domain of a cross product. Part of Theorem 3.13(x) of [Monk1]
p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmxpinm 4499* |
The domain of the intersection of two square cross products. Unlike
dmin 4486, equality holds. (Contributed by NM,
29-Jan-2008.)
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Theorem | xpid11m 4500* |
The cross product of a class with itself is one-to-one. (Contributed by
Jim Kingdon, 8-Dec-2018.)
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