Type  Label  Description 
Statement 

Theorem  ecopoveq 6201* 
This is the first of several theorems about equivalence relations of
the kind used in construction of fractions and signed reals, involving
operations on equivalent classes of ordered pairs. This theorem
expresses the relation (specified by the hypothesis) in terms
of its operation . (Contributed by NM, 16Aug1995.)



Theorem  ecopovsym 6202* 
Assuming the operation is commutative, show that the relation
,
specified by the first hypothesis, is symmetric.
(Contributed by NM, 27Aug1995.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  ecopovtrn 6203* 
Assuming that operation is commutative (second hypothesis),
closed (third hypothesis), associative (fourth hypothesis), and has
the cancellation property (fifth hypothesis), show that the relation
,
specified by the first hypothesis, is transitive.
(Contributed by NM, 11Feb1996.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  ecopover 6204* 
Assuming that operation is commutative (second hypothesis),
closed (third hypothesis), associative (fourth hypothesis), and has
the cancellation property (fifth hypothesis), show that the relation
,
specified by the first hypothesis, is an equivalence
relation. (Contributed by NM, 16Feb1996.) (Revised by Mario
Carneiro, 12Aug2015.)



Theorem  ecopovsymg 6205* 
Assuming the operation is commutative, show that the relation
,
specified by the first hypothesis, is symmetric.
(Contributed by Jim Kingdon, 1Sep2019.)



Theorem  ecopovtrng 6206* 
Assuming that operation is commutative (second hypothesis),
closed (third hypothesis), associative (fourth hypothesis), and has
the cancellation property (fifth hypothesis), show that the relation
,
specified by the first hypothesis, is transitive.
(Contributed by Jim Kingdon, 1Sep2019.)



Theorem  ecopoverg 6207* 
Assuming that operation is commutative (second hypothesis),
closed (third hypothesis), associative (fourth hypothesis), and has
the cancellation property (fifth hypothesis), show that the relation
,
specified by the first hypothesis, is an equivalence
relation. (Contributed by Jim Kingdon, 1Sep2019.)



Theorem  th3qlem1 6208* 
Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The
third hypothesis is the compatibility assumption. (Contributed by NM,
3Aug1995.) (Revised by Mario Carneiro, 9Jul2014.)



Theorem  th3qlem2 6209* 
Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60,
extended to operations on ordered pairs. The fourth hypothesis is the
compatibility assumption. (Contributed by NM, 4Aug1995.) (Revised by
Mario Carneiro, 12Aug2015.)



Theorem  th3qcor 6210* 
Corollary of Theorem 3Q of [Enderton] p. 60.
(Contributed by NM,
12Nov1995.) (Revised by David Abernethy, 4Jun2013.)



Theorem  th3q 6211* 
Theorem 3Q of [Enderton] p. 60, extended to
operations on ordered
pairs. (Contributed by NM, 4Aug1995.) (Revised by Mario Carneiro,
19Dec2013.)



Theorem  oviec 6212* 
Express an operation on equivalence classes of ordered pairs in terms of
equivalence class of operations on ordered pairs. See iset.mm for
additional comments describing the hypotheses. (Unnecessary distinct
variable restrictions were removed by David Abernethy, 4Jun2013.)
(Contributed by NM, 6Aug1995.) (Revised by Mario Carneiro,
4Jun2013.)



Theorem  ecovcom 6213* 
Lemma used to transfer a commutative law via an equivalence relation.
Most uses will want ecovicom 6214 instead. (Contributed by NM,
29Aug1995.) (Revised by David Abernethy, 4Jun2013.)



Theorem  ecovicom 6214* 
Lemma used to transfer a commutative law via an equivalence relation.
(Contributed by Jim Kingdon, 15Sep2019.)



Theorem  ecovass 6215* 
Lemma used to transfer an associative law via an equivalence relation.
In most cases ecoviass 6216 will be more useful. (Contributed by NM,
31Aug1995.) (Revised by David Abernethy, 4Jun2013.)



Theorem  ecoviass 6216* 
Lemma used to transfer an associative law via an equivalence relation.
(Contributed by Jim Kingdon, 16Sep2019.)



Theorem  ecovdi 6217* 
Lemma used to transfer a distributive law via an equivalence relation.
Most likely ecovidi 6218 will be more helpful. (Contributed by NM,
2Sep1995.) (Revised by David Abernethy, 4Jun2013.)



Theorem  ecovidi 6218* 
Lemma used to transfer a distributive law via an equivalence relation.
(Contributed by Jim Kingdon, 17Sep2019.)



2.6.25 Equinumerosity


Syntax  cen 6219 
Extend class definition to include the equinumerosity relation
("approximately equals" symbol)



Syntax  cdom 6220 
Extend class definition to include the dominance relation (curly
lessthanorequal)



Syntax  cfn 6221 
Extend class definition to include the class of all finite sets.



Definition  dfen 6222* 
Define the equinumerosity relation. Definition of [Enderton] p. 129.
We define
to be a binary relation rather than a connective, so
its arguments must be sets to be meaningful. This is acceptable because
we do not consider equinumerosity for proper classes. We derive the
usual definition as bren 6228. (Contributed by NM, 28Mar1998.)



Definition  dfdom 6223* 
Define the dominance relation. Compare Definition of [Enderton] p. 145.
Typical textbook definitions are derived as brdom 6231 and domen 6232.
(Contributed by NM, 28Mar1998.)



Definition  dffin 6224* 
Define the (proper) class of all finite sets. Similar to Definition
10.29 of [TakeutiZaring] p. 91,
whose "Fin(a)" corresponds to
our " ". This definition is
meaningful whether or not we
accept the Axiom of Infinity axinf2 10101. (Contributed by NM,
22Aug2008.)



Theorem  relen 6225 
Equinumerosity is a relation. (Contributed by NM, 28Mar1998.)



Theorem  reldom 6226 
Dominance is a relation. (Contributed by NM, 28Mar1998.)



Theorem  encv 6227 
If two classes are equinumerous, both classes are sets. (Contributed by
AV, 21Mar2019.)



Theorem  bren 6228* 
Equinumerosity relation. (Contributed by NM, 15Jun1998.)



Theorem  brdomg 6229* 
Dominance relation. (Contributed by NM, 15Jun1998.)



Theorem  brdomi 6230* 
Dominance relation. (Contributed by Mario Carneiro, 26Apr2015.)



Theorem  brdom 6231* 
Dominance relation. (Contributed by NM, 15Jun1998.)



Theorem  domen 6232* 
Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
(Contributed by NM, 15Jun1998.)



Theorem  domeng 6233* 
Dominance in terms of equinumerosity, with the sethood requirement
expressed as an antecedent. Example 1 of [Enderton] p. 146.
(Contributed by NM, 24Apr2004.)



Theorem  f1oen3g 6234 
The domain and range of a onetoone, onto function are equinumerous.
This variation of f1oeng 6237 does not require the Axiom of Replacement.
(Contributed by NM, 13Jan2007.) (Revised by Mario Carneiro,
10Sep2015.)



Theorem  f1oen2g 6235 
The domain and range of a onetoone, onto function are equinumerous.
This variation of f1oeng 6237 does not require the Axiom of Replacement.
(Contributed by Mario Carneiro, 10Sep2015.)



Theorem  f1dom2g 6236 
The domain of a onetoone function is dominated by its codomain. This
variation of f1domg 6238 does not require the Axiom of Replacement.
(Contributed by Mario Carneiro, 24Jun2015.)



Theorem  f1oeng 6237 
The domain and range of a onetoone, onto function are equinumerous.
(Contributed by NM, 19Jun1998.)



Theorem  f1domg 6238 
The domain of a onetoone function is dominated by its codomain.
(Contributed by NM, 4Sep2004.)



Theorem  f1oen 6239 
The domain and range of a onetoone, onto function are equinumerous.
(Contributed by NM, 19Jun1998.)



Theorem  f1dom 6240 
The domain of a onetoone function is dominated by its codomain.
(Contributed by NM, 19Jun1998.)



Theorem  isfi 6241* 
Express " is
finite." Definition 10.29 of [TakeutiZaring] p. 91
(whose " " is a predicate instead of a class). (Contributed by
NM, 22Aug2008.)



Theorem  enssdom 6242 
Equinumerosity implies dominance. (Contributed by NM, 31Mar1998.)



Theorem  endom 6243 
Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94.
(Contributed by NM, 28May1998.)



Theorem  enrefg 6244 
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 18Jun1998.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  enref 6245 
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 25Sep2004.)



Theorem  eqeng 6246 
Equality implies equinumerosity. (Contributed by NM, 26Oct2003.)



Theorem  domrefg 6247 
Dominance is reflexive. (Contributed by NM, 18Jun1998.)



Theorem  en2d 6248* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 27Jul2004.) (Revised by Mario Carneiro,
12May2014.)



Theorem  en3d 6249* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 27Jul2004.) (Revised by Mario Carneiro,
12May2014.)



Theorem  en2i 6250* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 4Jan2004.)



Theorem  en3i 6251* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 19Jul2004.)



Theorem  dom2lem 6252* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. (Contributed by NM,
24Jul2004.)



Theorem  dom2d 6253* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. (Contributed by NM,
24Jul2004.) (Revised by Mario Carneiro, 20May2013.)



Theorem  dom3d 6254* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. (Contributed by Mario
Carneiro, 20May2013.)



Theorem  dom2 6255* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. and can be
read and , as can be inferred from their
distinct variable conditions. (Contributed by NM, 26Oct2003.)



Theorem  dom3 6256* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. and can be
read and , as can be inferred from their
distinct variable conditions. (Contributed by Mario Carneiro,
20May2013.)



Theorem  idssen 6257 
Equality implies equinumerosity. (Contributed by NM, 30Apr1998.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  ssdomg 6258 
A set dominates its subsets. Theorem 16 of [Suppes] p. 94.
(Contributed by NM, 19Jun1998.) (Revised by Mario Carneiro,
24Jun2015.)



Theorem  ener 6259 
Equinumerosity is an equivalence relation. (Contributed by NM,
19Mar1998.) (Revised by Mario Carneiro, 15Nov2014.)



Theorem  ensymb 6260 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
Mario Carneiro, 26Apr2015.)



Theorem  ensym 6261 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
NM, 26Oct2003.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  ensymi 6262 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed
by NM, 25Sep2004.)



Theorem  ensymd 6263 
Symmetry of equinumerosity. Deduction form of ensym 6261. (Contributed
by David Moews, 1May2017.)



Theorem  entr 6264 
Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92.
(Contributed by NM, 9Jun1998.)



Theorem  domtr 6265 
Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94.
(Contributed by NM, 4Jun1998.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  entri 6266 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr2i 6267 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr3i 6268 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr4i 6269 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  endomtr 6270 
Transitivity of equinumerosity and dominance. (Contributed by NM,
7Jun1998.)



Theorem  domentr 6271 
Transitivity of dominance and equinumerosity. (Contributed by NM,
7Jun1998.)



Theorem  f1imaeng 6272 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (Contributed by Mario Carneiro, 15May2015.)



Theorem  f1imaen2g 6273 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (This version of f1imaen 6274 does not need axsetind 4262.)
(Contributed by Mario Carneiro, 16Nov2014.) (Revised by Mario Carneiro,
25Jun2015.)



Theorem  f1imaen 6274 
A onetoone function's image under a subset of its domain is
equinumerous to the subset. (Contributed by NM, 30Sep2004.)



Theorem  en0 6275 
The empty set is equinumerous only to itself. Exercise 1 of
[TakeutiZaring] p. 88.
(Contributed by NM, 27May1998.)



Theorem  ensn1 6276 
A singleton is equinumerous to ordinal one. (Contributed by NM,
4Nov2002.)



Theorem  ensn1g 6277 
A singleton is equinumerous to ordinal one. (Contributed by NM,
23Apr2004.)



Theorem  enpr1g 6278 
has only
one element. (Contributed by FL, 15Feb2010.)



Theorem  en1 6279* 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by NM, 25Jul2004.)



Theorem  en1bg 6280 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  reuen1 6281* 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1 6282 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1b 6283* 
Two ways to express " has a unique element". (Contributed by
Mario Carneiro, 9Apr2015.)



Theorem  en1uniel 6284 
A singleton contains its sole element. (Contributed by Stefan O'Rear,
16Aug2015.)



Theorem  2dom 6285* 
A set that dominates ordinal 2 has at least 2 different members.
(Contributed by NM, 25Jul2004.)



Theorem  fundmen 6286 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 28Jul2004.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  fundmeng 6287 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 17Sep2013.)



Theorem  cnven 6288 
A relational set is equinumerous to its converse. (Contributed by Mario
Carneiro, 28Dec2014.)



Theorem  fndmeng 6289 
A function is equinumerate to its domain. (Contributed by Paul Chapman,
22Jun2011.)



Theorem  en2sn 6290 
Two singletons are equinumerous. (Contributed by NM, 9Nov2003.)



Theorem  snfig 6291 
A singleton is finite. (Contributed by Jim Kingdon, 13Apr2020.)



Theorem  fiprc 6292 
The class of finite sets is a proper class. (Contributed by Jeff
Hankins, 3Oct2008.)



Theorem  unen 6293 
Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
(Contributed by NM, 11Jun1998.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  enm 6294* 
A set equinumerous to an inhabited set is inhabited. (Contributed by
Jim Kingdon, 19May2020.)



Theorem  xpsnen 6295 
A set is equinumerous to its Cartesian product with a singleton.
Proposition 4.22(c) of [Mendelson] p.
254. (Contributed by NM,
4Jan2004.) (Revised by Mario Carneiro, 15Nov2014.)



Theorem  xpsneng 6296 
A set is equinumerous to its Cartesian product with a singleton.
Proposition 4.22(c) of [Mendelson] p.
254. (Contributed by NM,
22Oct2004.)



Theorem  xp1en 6297 
One times a cardinal number. (Contributed by NM, 27Sep2004.) (Revised
by Mario Carneiro, 29Apr2015.)



Theorem  endisj 6298* 
Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
[Mendelson] p. 255. (Contributed by
NM, 16Apr2004.)



Theorem  xpcomf1o 6299* 
The canonical bijection from to .
(Contributed by Mario Carneiro, 23Apr2014.)



Theorem  xpcomco 6300* 
Composition with the bijection of xpcomf1o 6299 swaps the arguments to a
mapping. (Contributed by Mario Carneiro, 30May2015.)

