Type  Label  Description 
Statement 

Theorem  nn0eln0 4501 
A natural number is nonempty iff it contains the empty set. Although in
constructive mathematics it is generally more natural to work with
inhabited sets and ignore the whole concept of nonempty sets, in the
specific case of natural numbers this theorem may be helpful in converting
proofs which were written assuming excluded middle. (Contributed by Jim
Kingdon, 28Aug2019.)



Theorem  nnregexmid 4502* 
If inhabited sets of natural numbers always have minimal elements,
excluded middle follows. The argument is essentially the same as
regexmid 4418 and the larger lesson is that although
natural numbers may
behave "nonconstructively" even in a constructive set theory
(for
example see nndceq 6361 or nntri3or 6355), sets of natural numbers are a
different animal. (Contributed by Jim Kingdon, 6Sep2019.)



Theorem  omsinds 4503* 
Strong (or "total") induction principle over . (Contributed by
Scott Fenton, 17Jul2015.)



Theorem  nnpredcl 4504 
The predecessor of a natural number is a natural number. This theorem
is most interesting when the natural number is a successor (as seen in
theorems like onsucuni2 4447) but also holds when it is by
uni0 3731. (Contributed by Jim Kingdon, 31Jul2022.)



2.6.6 Relations


Syntax  cxp 4505 
Extend the definition of a class to include the cross product.



Syntax  ccnv 4506 
Extend the definition of a class to include the converse of a class.



Syntax  cdm 4507 
Extend the definition of a class to include the domain of a class.



Syntax  crn 4508 
Extend the definition of a class to include the range of a class.



Syntax  cres 4509 
Extend the definition of a class to include the restriction of a class.
(Read: The restriction of to .)



Syntax  cima 4510 
Extend the definition of a class to include the image of a class. (Read:
The image of
under .)



Syntax  ccom 4511 
Extend the definition of a class to include the composition of two
classes. (Read: The composition of and .)



Syntax  wrel 4512 
Extend the definition of a wff to include the relation predicate. (Read:
is a relation.)



Definition  dfxp 4513* 
Define the cross product of two classes. Definition 9.11 of [Quine]
p. 64. For example, ( { 1 , 5 } { 2 , 7 } ) = ( { 1 , 2
,
1 , 7 } { 5 , 2 , 5 ,
7 }
) . Another example is that the set of rational numbers are
defined in using the crossproduct ( Z N ) ; the left and
righthand sides of the crossproduct represent the top (integer) and
bottom (natural) numbers of a fraction. (Contributed by NM,
4Jul1994.)



Definition  dfrel 4514 
Define the relation predicate. Definition 6.4(1) of [TakeutiZaring]
p. 23. For alternate definitions, see dfrel2 4957 and dfrel3 4964.
(Contributed by NM, 1Aug1994.)



Definition  dfcnv 4515* 
Define the converse of a class. Definition 9.12 of [Quine] p. 64. The
converse of a binary relation swaps its arguments, i.e., if
and then , as proven in brcnv 4690
(see dfbr 3898 and dfrel 4514 for more on relations). For example,
{ 2
, 6 , 3 , 9 } = { 6 , 2 ,
9 ,
3 } . We
use Quine's breve accent (smile) notation.
Like Quine, we use it as a prefix, which eliminates the need for
parentheses. Many authors use the postfix superscript "to the
minus
one." "Converse" is Quine's terminology; some authors
call it
"inverse," especially when the argument is a function.
(Contributed by
NM, 4Jul1994.)



Definition  dfco 4516* 
Define the composition of two classes. Definition 6.6(3) of
[TakeutiZaring] p. 24. Note that
Definition 7 of [Suppes] p. 63
reverses and
, uses a slash instead
of , and calls
the operation "relative product." (Contributed by NM,
4Jul1994.)



Definition  dfdm 4517* 
Define the domain of a class. Definition 3 of [Suppes] p. 59. For
example, F = { 2 , 6 , 3 , 9 } dom F
= { 2 , 3 } . Contrast with range (defined in dfrn 4518). For alternate
definitions see dfdm2 5041, dfdm3 4694, and dfdm4 4699. The
notation " " is used by Enderton; other authors sometimes use
script D. (Contributed by NM, 1Aug1994.)



Definition  dfrn 4518 
Define the range of a class. For example, F = { 2 , 6 ,
3 ,
9 } >
ran F = { 6 , 9 } . Contrast with domain
(defined in dfdm 4517). For alternate definitions, see dfrn2 4695,
dfrn3 4696, and dfrn4 4967. The notation " " is used by Enderton;
other authors sometimes use script R or script W. (Contributed by NM,
1Aug1994.)



Definition  dfres 4519 
Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring]
p. 24. For example ( F = { 2 , 6 , 3 , 9 }
B = { 1 ,
2 } ) > ( F B ) = { 2 , 6 } .
(Contributed by NM, 2Aug1994.)



Definition  dfima 4520 
Define the image of a class (as restricted by another class).
Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = {
2 , 6 , 3 , 9 } /\ B = { 1 , 2 } ) > ( F B )
= { 6 } . Contrast with restriction (dfres 4519) and range (dfrn 4518).
For an alternate definition, see dfima2 4851. (Contributed by NM,
2Aug1994.)



Theorem  xpeq1 4521 
Equality theorem for cross product. (Contributed by NM, 4Jul1994.)



Theorem  xpeq2 4522 
Equality theorem for cross product. (Contributed by NM, 5Jul1994.)



Theorem  elxpi 4523* 
Membership in a cross product. Uses fewer axioms than elxp 4524.
(Contributed by NM, 4Jul1994.)



Theorem  elxp 4524* 
Membership in a cross product. (Contributed by NM, 4Jul1994.)



Theorem  elxp2 4525* 
Membership in a cross product. (Contributed by NM, 23Feb2004.)



Theorem  xpeq12 4526 
Equality theorem for cross product. (Contributed by FL, 31Aug2009.)



Theorem  xpeq1i 4527 
Equality inference for cross product. (Contributed by NM,
21Dec2008.)



Theorem  xpeq2i 4528 
Equality inference for cross product. (Contributed by NM,
21Dec2008.)



Theorem  xpeq12i 4529 
Equality inference for cross product. (Contributed by FL,
31Aug2009.)



Theorem  xpeq1d 4530 
Equality deduction for cross product. (Contributed by Jeff Madsen,
17Jun2010.)



Theorem  xpeq2d 4531 
Equality deduction for cross product. (Contributed by Jeff Madsen,
17Jun2010.)



Theorem  xpeq12d 4532 
Equality deduction for Cartesian product. (Contributed by NM,
8Dec2013.)



Theorem  sqxpeqd 4533 
Equality deduction for a Cartesian square, see Wikipedia "Cartesian
product",
https://en.wikipedia.org/wiki/Cartesian_product#nary_Cartesian_power.
(Contributed by AV, 13Jan2020.)



Theorem  nfxp 4534 
Boundvariable hypothesis builder for cross product. (Contributed by
NM, 15Sep2003.) (Revised by Mario Carneiro, 15Oct2016.)



Theorem  0nelxp 4535 
The empty set is not a member of a cross product. (Contributed by NM,
2May1996.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  0nelelxp 4536 
A member of a cross product (ordered pair) doesn't contain the empty
set. (Contributed by NM, 15Dec2008.)



Theorem  opelxp 4537 
Ordered pair membership in a cross product. (Contributed by NM,
15Nov1994.) (Proof shortened by Andrew Salmon, 12Aug2011.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  brxp 4538 
Binary relation on a cross product. (Contributed by NM,
22Apr2004.)



Theorem  opelxpi 4539 
Ordered pair membership in a cross product (implication). (Contributed by
NM, 28May1995.)



Theorem  opelxpd 4540 
Ordered pair membership in a Cartesian product, deduction form.
(Contributed by Glauco Siliprandi, 3Mar2021.)



Theorem  opelxp1 4541 
The first member of an ordered pair of classes in a cross product belongs
to first cross product argument. (Contributed by NM, 28May2008.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  opelxp2 4542 
The second member of an ordered pair of classes in a cross product belongs
to second cross product argument. (Contributed by Mario Carneiro,
26Apr2015.)



Theorem  otelxp1 4543 
The first member of an ordered triple of classes in a cross product
belongs to first cross product argument. (Contributed by NM,
28May2008.)



Theorem  rabxp 4544* 
Membership in a class builder restricted to a cross product.
(Contributed by NM, 20Feb2014.)



Theorem  brrelex12 4545 
A true binary relation on a relation implies the arguments are sets.
(This is a property of our ordered pair definition.) (Contributed by
Mario Carneiro, 26Apr2015.)



Theorem  brrelex1 4546 
A true binary relation on a relation implies the first argument is a set.
(This is a property of our ordered pair definition.) (Contributed by NM,
18May2004.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  brrelex 4547 
A true binary relation on a relation implies the first argument is a set.
(This is a property of our ordered pair definition.) (Contributed by NM,
18May2004.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  brrelex2 4548 
A true binary relation on a relation implies the second argument is a set.
(This is a property of our ordered pair definition.) (Contributed by
Mario Carneiro, 26Apr2015.)



Theorem  brrelex12i 4549 
Two classes that are related by a binary relation are sets. (An
artifact of our ordered pair definition.) (Contributed by BJ,
3Oct2022.)



Theorem  brrelex1i 4550 
The first argument of a binary relation exists. (An artifact of our
ordered pair definition.) (Contributed by NM, 4Jun1998.)



Theorem  brrelex2i 4551 
The second argument of a binary relation exists. (An artifact of our
ordered pair definition.) (Contributed by Mario Carneiro,
26Apr2015.)



Theorem  nprrel 4552 
No proper class is related to anything via any relation. (Contributed
by Roy F. Longton, 30Jul2005.)



Theorem  0nelrel 4553 
A binary relation does not contain the empty set. (Contributed by AV,
15Nov2021.)



Theorem  fconstmpt 4554* 
Representation of a constant function using the mapping operation.
(Note that
cannot appear free in .) (Contributed by NM,
12Oct1999.) (Revised by Mario Carneiro, 16Nov2013.)



Theorem  vtoclr 4555* 
Variable to class conversion of transitive relation. (Contributed by
NM, 9Jun1998.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  opelvvg 4556 
Ordered pair membership in the universal class of ordered pairs.
(Contributed by Mario Carneiro, 3May2015.)



Theorem  opelvv 4557 
Ordered pair membership in the universal class of ordered pairs.
(Contributed by NM, 22Aug2013.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  opthprc 4558 
Justification theorem for an ordered pair definition that works for any
classes, including proper classes. This is a possible definition
implied by the footnote in [Jech] p. 78,
which says, "The sophisticated
reader will not object to our use of a pair of classes."
(Contributed
by NM, 28Sep2003.)



Theorem  brel 4559 
Two things in a binary relation belong to the relation's domain.
(Contributed by NM, 17May1996.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  brab2a 4560* 
Ordered pair membership in an ordered pair class abstraction.
(Contributed by Mario Carneiro, 9Nov2015.)



Theorem  elxp3 4561* 
Membership in a cross product. (Contributed by NM, 5Mar1995.)



Theorem  opeliunxp 4562 
Membership in a union of cross products. (Contributed by Mario
Carneiro, 29Dec2014.) (Revised by Mario Carneiro, 1Jan2017.)



Theorem  xpundi 4563 
Distributive law for cross product over union. Theorem 103 of [Suppes]
p. 52. (Contributed by NM, 12Aug2004.)



Theorem  xpundir 4564 
Distributive law for cross product over union. Similar to Theorem 103
of [Suppes] p. 52. (Contributed by NM,
30Sep2002.)



Theorem  xpiundi 4565* 
Distributive law for cross product over indexed union. (Contributed by
Mario Carneiro, 27Apr2014.)



Theorem  xpiundir 4566* 
Distributive law for cross product over indexed union. (Contributed by
Mario Carneiro, 27Apr2014.)



Theorem  iunxpconst 4567* 
Membership in a union of cross products when the second factor is
constant. (Contributed by Mario Carneiro, 29Dec2014.)



Theorem  xpun 4568 
The cross product of two unions. (Contributed by NM, 12Aug2004.)



Theorem  elvv 4569* 
Membership in universal class of ordered pairs. (Contributed by NM,
4Jul1994.)



Theorem  elvvv 4570* 
Membership in universal class of ordered triples. (Contributed by NM,
17Dec2008.)



Theorem  elvvuni 4571 
An ordered pair contains its union. (Contributed by NM,
16Sep2006.)



Theorem  mosubopt 4572* 
"At most one" remains true inside ordered pair quantification.
(Contributed by NM, 28Aug2007.)



Theorem  mosubop 4573* 
"At most one" remains true inside ordered pair quantification.
(Contributed by NM, 28May1995.)



Theorem  brinxp2 4574 
Intersection of binary relation with cross product. (Contributed by NM,
3Mar2007.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  brinxp 4575 
Intersection of binary relation with cross product. (Contributed by NM,
9Mar1997.)



Theorem  poinxp 4576 
Intersection of partial order with cross product of its field.
(Contributed by Mario Carneiro, 10Jul2014.)



Theorem  soinxp 4577 
Intersection of linear order with cross product of its field.
(Contributed by Mario Carneiro, 10Jul2014.)



Theorem  seinxp 4578 
Intersection of setlike relation with cross product of its field.
(Contributed by Mario Carneiro, 22Jun2015.)

Se
Se 

Theorem  posng 4579 
Partial ordering of a singleton. (Contributed by Jim Kingdon,
5Dec2018.)



Theorem  sosng 4580 
Strict linear ordering on a singleton. (Contributed by Jim Kingdon,
5Dec2018.)



Theorem  opabssxp 4581* 
An abstraction relation is a subset of a related cross product.
(Contributed by NM, 16Jul1995.)



Theorem  brab2ga 4582* 
The law of concretion for a binary relation. See brab2a 4560 for alternate
proof. TODO: should one of them be deleted? (Contributed by Mario
Carneiro, 28Apr2015.) (Proof modification is discouraged.)



Theorem  optocl 4583* 
Implicit substitution of class for ordered pair. (Contributed by NM,
5Mar1995.)



Theorem  2optocl 4584* 
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12Mar1995.)



Theorem  3optocl 4585* 
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12Mar1995.)



Theorem  opbrop 4586* 
Ordered pair membership in a relation. Special case. (Contributed by
NM, 5Aug1995.)



Theorem  0xp 4587 
The cross product with the empty set is empty. Part of Theorem 3.13(ii)
of [Monk1] p. 37. (Contributed by NM,
4Jul1994.)



Theorem  csbxpg 4588 
Distribute proper substitution through the cross product of two classes.
(Contributed by Alan Sare, 10Nov2012.)



Theorem  releq 4589 
Equality theorem for the relation predicate. (Contributed by NM,
1Aug1994.)



Theorem  releqi 4590 
Equality inference for the relation predicate. (Contributed by NM,
8Dec2006.)



Theorem  releqd 4591 
Equality deduction for the relation predicate. (Contributed by NM,
8Mar2014.)



Theorem  nfrel 4592 
Boundvariable hypothesis builder for a relation. (Contributed by NM,
31Jan2004.) (Revised by Mario Carneiro, 15Oct2016.)



Theorem  sbcrel 4593 
Distribute proper substitution through a relation predicate. (Contributed
by Alexander van der Vekens, 23Jul2017.)



Theorem  relss 4594 
Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
(Contributed by NM, 15Aug1994.)



Theorem  ssrel 4595* 
A subclass relationship depends only on a relation's ordered pairs.
Theorem 3.2(i) of [Monk1] p. 33.
(Contributed by NM, 2Aug1994.)
(Proof shortened by Andrew Salmon, 27Aug2011.)



Theorem  eqrel 4596* 
Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
p. 33. (Contributed by NM, 2Aug1994.)



Theorem  ssrel2 4597* 
A subclass relationship depends only on a relation's ordered pairs.
This version of ssrel 4595 is restricted to the relation's domain.
(Contributed by Thierry Arnoux, 25Jan2018.)



Theorem  relssi 4598* 
Inference from subclass principle for relations. (Contributed by NM,
31Mar1998.)



Theorem  relssdv 4599* 
Deduction from subclass principle for relations. (Contributed by NM,
11Sep2004.)



Theorem  eqrelriv 4600* 
Inference from extensionality principle for relations. (Contributed by
FL, 15Oct2012.)

