Theorem List for Intuitionistic Logic Explorer - 4901-5000 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | resieq 4901 |
A restricted identity relation is equivalent to equality in its domain.
(Contributed by NM, 30-Apr-2004.)
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Theorem | opelresi 4902 |
belongs to a restriction of the identity class iff
belongs to the restricting class. (Contributed by FL, 27-Oct-2008.)
(Revised by NM, 30-Mar-2016.)
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Theorem | resres 4903 |
The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
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Theorem | resundi 4904 |
Distributive law for restriction over union. Theorem 31 of [Suppes]
p. 65. (Contributed by NM, 30-Sep-2002.)
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Theorem | resundir 4905 |
Distributive law for restriction over union. (Contributed by NM,
23-Sep-2004.)
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Theorem | resindi 4906 |
Class restriction distributes over intersection. (Contributed by FL,
6-Oct-2008.)
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Theorem | resindir 4907 |
Class restriction distributes over intersection. (Contributed by NM,
18-Dec-2008.)
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Theorem | inres 4908 |
Move intersection into class restriction. (Contributed by NM,
18-Dec-2008.)
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Theorem | resdifcom 4909 |
Commutative law for restriction and difference. (Contributed by AV,
7-Jun-2021.)
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Theorem | resiun1 4910* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
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Theorem | resiun2 4911* |
Distribution of restriction over indexed union. (Contributed by Mario
Carneiro, 29-May-2015.)
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Theorem | dmres 4912 |
The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
(Contributed by NM, 1-Aug-1994.)
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Theorem | ssdmres 4913 |
A domain restricted to a subclass equals the subclass. (Contributed by
NM, 2-Mar-1997.)
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Theorem | dmresexg 4914 |
The domain of a restriction to a set exists. (Contributed by NM,
7-Apr-1995.)
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Theorem | resss 4915 |
A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.)
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Theorem | rescom 4916 |
Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | ssres 4917 |
Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
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Theorem | ssres2 4918 |
Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | relres 4919 |
A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
(Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | resabs1 4920 |
Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
(Contributed by NM, 9-Aug-1994.)
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Theorem | resabs1d 4921 |
Absorption law for restriction, deduction form. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
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Theorem | resabs2 4922 |
Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | residm 4923 |
Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
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Theorem | resima 4924 |
A restriction to an image. (Contributed by NM, 29-Sep-2004.)
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Theorem | resima2 4925 |
Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
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Theorem | xpssres 4926 |
Restriction of a constant function (or other cross product). (Contributed
by Stefan O'Rear, 24-Jan-2015.)
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Theorem | elres 4927* |
Membership in a restriction. (Contributed by Scott Fenton,
17-Mar-2011.)
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Theorem | elsnres 4928* |
Memebership in restriction to a singleton. (Contributed by Scott
Fenton, 17-Mar-2011.)
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Theorem | relssres 4929 |
Simplification law for restriction. (Contributed by NM,
16-Aug-1994.)
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Theorem | resdm 4930 |
A relation restricted to its domain equals itself. (Contributed by NM,
12-Dec-2006.)
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Theorem | resexg 4931 |
The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | resex 4932 |
The restriction of a set is a set. (Contributed by Jeff Madsen,
19-Jun-2011.)
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Theorem | resindm 4933 |
When restricting a relation, intersecting with the domain of the relation
has no effect. (Contributed by FL, 6-Oct-2008.)
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Theorem | resdmdfsn 4934 |
Restricting a relation to its domain without a set is the same as
restricting the relation to the universe without this set. (Contributed
by AV, 2-Dec-2018.)
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Theorem | resopab 4935* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 5-Nov-2002.)
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Theorem | resiexg 4936 |
The existence of a restricted identity function, proved without using
the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
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Theorem | iss 4937 |
A subclass of the identity function is the identity function restricted
to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | resopab2 4938* |
Restriction of a class abstraction of ordered pairs. (Contributed by
NM, 24-Aug-2007.)
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Theorem | resmpt 4939* |
Restriction of the mapping operation. (Contributed by Mario Carneiro,
15-Jul-2013.)
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Theorem | resmpt3 4940* |
Unconditional restriction of the mapping operation. (Contributed by
Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro,
22-Mar-2015.)
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Theorem | resmptf 4941 |
Restriction of the mapping operation. (Contributed by Thierry Arnoux,
28-Mar-2017.)
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Theorem | resmptd 4942* |
Restriction of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | dfres2 4943* |
Alternate definition of the restriction operation. (Contributed by
Mario Carneiro, 5-Nov-2013.)
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Theorem | opabresid 4944* |
The restricted identity expressed with the class builder. (Contributed
by FL, 25-Apr-2012.)
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Theorem | mptresid 4945* |
The restricted identity expressed with the maps-to notation.
(Contributed by FL, 25-Apr-2012.)
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Theorem | dmresi 4946 |
The domain of a restricted identity function. (Contributed by NM,
27-Aug-2004.)
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Theorem | resid 4947 |
Any relation restricted to the universe is itself. (Contributed by NM,
16-Mar-2004.)
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Theorem | imaeq1 4948 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
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Theorem | imaeq2 4949 |
Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
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Theorem | imaeq1i 4950 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
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Theorem | imaeq2i 4951 |
Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
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Theorem | imaeq1d 4952 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
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Theorem | imaeq2d 4953 |
Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
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Theorem | imaeq12d 4954 |
Equality theorem for image. (Contributed by Mario Carneiro,
4-Dec-2016.)
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Theorem | dfima2 4955* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dfima3 4956* |
Alternate definition of image. Compare definition (d) of [Enderton]
p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | elimag 4957* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 20-Jan-2007.)
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Theorem | elima 4958* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 19-Apr-2004.)
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Theorem | elima2 4959* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 11-Aug-2004.)
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Theorem | elima3 4960* |
Membership in an image. Theorem 34 of [Suppes]
p. 65. (Contributed by
NM, 14-Aug-1994.)
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Theorem | nfima 4961 |
Bound-variable hypothesis builder for image. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | nfimad 4962 |
Deduction version of bound-variable hypothesis builder nfima 4961.
(Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro,
15-Oct-2016.)
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Theorem | imadmrn 4963 |
The image of the domain of a class is the range of the class.
(Contributed by NM, 14-Aug-1994.)
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Theorem | imassrn 4964 |
The image of a class is a subset of its range. Theorem 3.16(xi) of
[Monk1] p. 39. (Contributed by NM,
31-Mar-1995.)
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Theorem | imaexg 4965 |
The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed
by NM, 24-Jul-1995.)
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Theorem | imaex 4966 |
The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
(Contributed by JJ, 24-Sep-2021.)
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Theorem | imai 4967 |
Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
(Contributed by NM, 30-Apr-1998.)
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Theorem | rnresi 4968 |
The range of the restricted identity function. (Contributed by NM,
27-Aug-2004.)
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Theorem | resiima 4969 |
The image of a restriction of the identity function. (Contributed by FL,
31-Dec-2006.)
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Theorem | ima0 4970 |
Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed
by NM, 20-May-1998.)
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Theorem | 0ima 4971 |
Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
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Theorem | csbima12g 4972 |
Move class substitution in and out of the image of a function.
(Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro,
4-Dec-2016.)
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Theorem | imadisj 4973 |
A class whose image under another is empty is disjoint with the other's
domain. (Contributed by FL, 24-Jan-2007.)
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Theorem | cnvimass 4974 |
A preimage under any class is included in the domain of the class.
(Contributed by FL, 29-Jan-2007.)
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Theorem | cnvimarndm 4975 |
The preimage of the range of a class is the domain of the class.
(Contributed by Jeff Hankins, 15-Jul-2009.)
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Theorem | imasng 4976* |
The image of a singleton. (Contributed by NM, 8-May-2005.)
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Theorem | elreimasng 4977 |
Elementhood in the image of a singleton. (Contributed by Jim Kingdon,
10-Dec-2018.)
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Theorem | elimasn 4978 |
Membership in an image of a singleton. (Contributed by NM,
15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | elimasng 4979 |
Membership in an image of a singleton. (Contributed by Raph Levien,
21-Oct-2006.)
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Theorem | args 4980* |
Two ways to express the class of unique-valued arguments of ,
which is the same as the domain of whenever is a function.
The left-hand side of the equality is from Definition 10.2 of [Quine]
p. 65. Quine uses the notation "arg " for this class (for which
we have no separate notation). (Contributed by NM, 8-May-2005.)
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Theorem | eliniseg 4981 |
Membership in an initial segment. The idiom ,
meaning , is used to specify an initial segment in
(for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | epini 4982 |
Any set is equal to its preimage under the converse epsilon relation.
(Contributed by Mario Carneiro, 9-Mar-2013.)
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Theorem | iniseg 4983* |
An idiom that signifies an initial segment of an ordering, used, for
example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by
NM, 28-Apr-2004.)
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Theorem | dfse2 4984* |
Alternate definition of set-like relation. (Contributed by Mario
Carneiro, 23-Jun-2015.)
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Theorem | exse2 4985 |
Any set relation is set-like. (Contributed by Mario Carneiro,
22-Jun-2015.)
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Theorem | imass1 4986 |
Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
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Theorem | imass2 4987 |
Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
(Contributed by NM, 22-Mar-1998.)
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Theorem | ndmima 4988 |
The image of a singleton outside the domain is empty. (Contributed by NM,
22-May-1998.)
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Theorem | relcnv 4989 |
A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed
by NM, 29-Oct-1996.)
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Theorem | relbrcnvg 4990 |
When is a relation,
the sethood assumptions on brcnv 4794 can be
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
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Theorem | relbrcnv 4991 |
When is a relation,
the sethood assumptions on brcnv 4794 can be
omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
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Theorem | cotr 4992* |
Two ways of saying a relation is transitive. Definition of transitivity
in [Schechter] p. 51. (Contributed by
NM, 27-Dec-1996.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | issref 4993* |
Two ways to state a relation is reflexive. Adapted from Tarski.
(Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
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Theorem | cnvsym 4994* |
Two ways of saying a relation is symmetric. Similar to definition of
symmetry in [Schechter] p. 51.
(Contributed by NM, 28-Dec-1996.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | intasym 4995* |
Two ways of saying a relation is antisymmetric. Definition of
antisymmetry in [Schechter] p. 51.
(Contributed by NM, 9-Sep-2004.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | asymref 4996* |
Two ways of saying a relation is antisymmetric and reflexive.
is the field of a relation by relfld 5139. (Contributed by
NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | intirr 4997* |
Two ways of saying a relation is irreflexive. Definition of
irreflexivity in [Schechter] p. 51.
(Contributed by NM, 9-Sep-2004.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | brcodir 4998* |
Two ways of saying that two elements have an upper bound. (Contributed
by Mario Carneiro, 3-Nov-2015.)
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Theorem | codir 4999* |
Two ways of saying a relation is directed. (Contributed by Mario
Carneiro, 22-Nov-2013.)
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Theorem | qfto 5000* |
A quantifier-free way of expressing the total order predicate.
(Contributed by Mario Carneiro, 22-Nov-2013.)
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