Type  Label  Description 
Statement 

Theorem  ixpm 6501* 
If an infinite Cartesian product of a family is inhabited,
every is inhabited. (Contributed by Mario Carneiro,
22Jun2016.) (Revised by Jim Kingdon, 16Feb2023.)



Theorem  ixp0 6502 
The infinite Cartesian product of a family with an empty
member is empty. (Contributed by NM, 1Oct2006.) (Revised by Jim
Kingdon, 16Feb2023.)



Theorem  ixpssmap 6503* 
An infinite Cartesian product is a subset of set exponentiation. Remark
in [Enderton] p. 54. (Contributed by
NM, 28Sep2006.)



Theorem  resixp 6504* 
Restriction of an element of an infinite Cartesian product.
(Contributed by FL, 7Nov2011.) (Proof shortened by Mario Carneiro,
31May2014.)



Theorem  mptelixpg 6505* 
Condition for an explicit member of an indexed product. (Contributed by
Stefan O'Rear, 4Jan2015.)



Theorem  elixpsn 6506* 
Membership in a class of singleton functions. (Contributed by Stefan
O'Rear, 24Jan2015.)



Theorem  ixpsnf1o 6507* 
A bijection between a class and singlepoint functions to it.
(Contributed by Stefan O'Rear, 24Jan2015.)



Theorem  mapsnf1o 6508* 
A bijection between a set and singlepoint functions to it.
(Contributed by Stefan O'Rear, 24Jan2015.)



2.6.27 Equinumerosity


Syntax  cen 6509 
Extend class definition to include the equinumerosity relation
("approximately equals" symbol)



Syntax  cdom 6510 
Extend class definition to include the dominance relation (curly
lessthanorequal)



Syntax  cfn 6511 
Extend class definition to include the class of all finite sets.



Definition  dfen 6512* 
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 6518. (Contributed by NM, 28Mar1998.)



Definition  dfdom 6513* 
Define the dominance relation. Compare Definition of [Enderton] p. 145.
Typical textbook definitions are derived as brdom 6521 and domen 6522.
(Contributed by NM, 28Mar1998.)



Definition  dffin 6514* 
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 12137. (Contributed by NM,
22Aug2008.)



Theorem  relen 6515 
Equinumerosity is a relation. (Contributed by NM, 28Mar1998.)



Theorem  reldom 6516 
Dominance is a relation. (Contributed by NM, 28Mar1998.)



Theorem  encv 6517 
If two classes are equinumerous, both classes are sets. (Contributed by
AV, 21Mar2019.)



Theorem  bren 6518* 
Equinumerosity relation. (Contributed by NM, 15Jun1998.)



Theorem  brdomg 6519* 
Dominance relation. (Contributed by NM, 15Jun1998.)



Theorem  brdomi 6520* 
Dominance relation. (Contributed by Mario Carneiro, 26Apr2015.)



Theorem  brdom 6521* 
Dominance relation. (Contributed by NM, 15Jun1998.)



Theorem  domen 6522* 
Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
(Contributed by NM, 15Jun1998.)



Theorem  domeng 6523* 
Dominance in terms of equinumerosity, with the sethood requirement
expressed as an antecedent. Example 1 of [Enderton] p. 146.
(Contributed by NM, 24Apr2004.)



Theorem  ctex 6524 
A countable set is a set. (Contributed by Thierry Arnoux,
29Dec2016.)



Theorem  f1oen3g 6525 
The domain and range of a onetoone, onto function are equinumerous.
This variation of f1oeng 6528 does not require the Axiom of Replacement.
(Contributed by NM, 13Jan2007.) (Revised by Mario Carneiro,
10Sep2015.)



Theorem  f1oen2g 6526 
The domain and range of a onetoone, onto function are equinumerous.
This variation of f1oeng 6528 does not require the Axiom of Replacement.
(Contributed by Mario Carneiro, 10Sep2015.)



Theorem  f1dom2g 6527 
The domain of a onetoone function is dominated by its codomain. This
variation of f1domg 6529 does not require the Axiom of Replacement.
(Contributed by Mario Carneiro, 24Jun2015.)



Theorem  f1oeng 6528 
The domain and range of a onetoone, onto function are equinumerous.
(Contributed by NM, 19Jun1998.)



Theorem  f1domg 6529 
The domain of a onetoone function is dominated by its codomain.
(Contributed by NM, 4Sep2004.)



Theorem  f1oen 6530 
The domain and range of a onetoone, onto function are equinumerous.
(Contributed by NM, 19Jun1998.)



Theorem  f1dom 6531 
The domain of a onetoone function is dominated by its codomain.
(Contributed by NM, 19Jun1998.)



Theorem  isfi 6532* 
Express " is
finite." Definition 10.29 of [TakeutiZaring] p. 91
(whose " " is a predicate instead of a class). (Contributed by
NM, 22Aug2008.)



Theorem  enssdom 6533 
Equinumerosity implies dominance. (Contributed by NM, 31Mar1998.)



Theorem  endom 6534 
Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94.
(Contributed by NM, 28May1998.)



Theorem  enrefg 6535 
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 18Jun1998.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  enref 6536 
Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed
by NM, 25Sep2004.)



Theorem  eqeng 6537 
Equality implies equinumerosity. (Contributed by NM, 26Oct2003.)



Theorem  domrefg 6538 
Dominance is reflexive. (Contributed by NM, 18Jun1998.)



Theorem  en2d 6539* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 27Jul2004.) (Revised by Mario Carneiro,
12May2014.)



Theorem  en3d 6540* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 27Jul2004.) (Revised by Mario Carneiro,
12May2014.)



Theorem  en2i 6541* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 4Jan2004.)



Theorem  en3i 6542* 
Equinumerosity inference from an implicit onetoone onto function.
(Contributed by NM, 19Jul2004.)



Theorem  dom2lem 6543* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. (Contributed by NM,
24Jul2004.)



Theorem  dom2d 6544* 
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 6545* 
A mapping (first hypothesis) that is onetoone (second hypothesis)
implies its domain is dominated by its codomain. (Contributed by Mario
Carneiro, 20May2013.)



Theorem  dom2 6546* 
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 6547* 
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 6548 
Equality implies equinumerosity. (Contributed by NM, 30Apr1998.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  ssdomg 6549 
A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed
by NM, 19Jun1998.) (Revised by Mario Carneiro, 24Jun2015.)



Theorem  ener 6550 
Equinumerosity is an equivalence relation. (Contributed by NM,
19Mar1998.) (Revised by Mario Carneiro, 15Nov2014.)



Theorem  ensymb 6551 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
Mario Carneiro, 26Apr2015.)



Theorem  ensym 6552 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
NM, 26Oct2003.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  ensymi 6553 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed
by NM, 25Sep2004.)



Theorem  ensymd 6554 
Symmetry of equinumerosity. Deduction form of ensym 6552. (Contributed
by David Moews, 1May2017.)



Theorem  entr 6555 
Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92.
(Contributed by NM, 9Jun1998.)



Theorem  domtr 6556 
Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94.
(Contributed by NM, 4Jun1998.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  entri 6557 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr2i 6558 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr3i 6559 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr4i 6560 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  endomtr 6561 
Transitivity of equinumerosity and dominance. (Contributed by NM,
7Jun1998.)



Theorem  domentr 6562 
Transitivity of dominance and equinumerosity. (Contributed by NM,
7Jun1998.)



Theorem  f1imaeng 6563 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (Contributed by Mario Carneiro, 15May2015.)



Theorem  f1imaen2g 6564 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (This version of f1imaen 6565 does not need axsetind 4366.)
(Contributed by Mario Carneiro, 16Nov2014.) (Revised by Mario Carneiro,
25Jun2015.)



Theorem  f1imaen 6565 
A onetoone function's image under a subset of its domain is
equinumerous to the subset. (Contributed by NM, 30Sep2004.)



Theorem  en0 6566 
The empty set is equinumerous only to itself. Exercise 1 of
[TakeutiZaring] p. 88.
(Contributed by NM, 27May1998.)



Theorem  ensn1 6567 
A singleton is equinumerous to ordinal one. (Contributed by NM,
4Nov2002.)



Theorem  ensn1g 6568 
A singleton is equinumerous to ordinal one. (Contributed by NM,
23Apr2004.)



Theorem  enpr1g 6569 
has only
one element. (Contributed by FL, 15Feb2010.)



Theorem  en1 6570* 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by NM, 25Jul2004.)



Theorem  en1bg 6571 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  reuen1 6572* 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1 6573 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1b 6574* 
Two ways to express " has a unique element". (Contributed by
Mario Carneiro, 9Apr2015.)



Theorem  en1uniel 6575 
A singleton contains its sole element. (Contributed by Stefan O'Rear,
16Aug2015.)



Theorem  2dom 6576* 
A set that dominates ordinal 2 has at least 2 different members.
(Contributed by NM, 25Jul2004.)



Theorem  fundmen 6577 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 28Jul2004.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  fundmeng 6578 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 17Sep2013.)



Theorem  cnven 6579 
A relational set is equinumerous to its converse. (Contributed by Mario
Carneiro, 28Dec2014.)



Theorem  cnvct 6580 
If a set is countable, so is its converse. (Contributed by Thierry
Arnoux, 29Dec2016.)



Theorem  fndmeng 6581 
A function is equinumerate to its domain. (Contributed by Paul Chapman,
22Jun2011.)



Theorem  mapsnen 6582 
Set exponentiation to a singleton exponent is equinumerous to its base.
Exercise 4.43 of [Mendelson] p. 255.
(Contributed by NM, 17Dec2003.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  map1 6583 
Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1.
Exercise 4.42(b) of [Mendelson] p.
255. (Contributed by NM,
17Dec2003.)



Theorem  en2sn 6584 
Two singletons are equinumerous. (Contributed by NM, 9Nov2003.)



Theorem  snfig 6585 
A singleton is finite. For the proper class case, see snprc 3511.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  fiprc 6586 
The class of finite sets is a proper class. (Contributed by Jeff
Hankins, 3Oct2008.)



Theorem  unen 6587 
Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
(Contributed by NM, 11Jun1998.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  ssct 6588 
Any subset of a countable set is countable. (Contributed by Thierry
Arnoux, 31Jan2017.)



Theorem  1domsn 6589 
A singleton (whether of a set or a proper class) is dominated by one.
(Contributed by Jim Kingdon, 1Mar2022.)



Theorem  enm 6590* 
A set equinumerous to an inhabited set is inhabited. (Contributed by
Jim Kingdon, 19May2020.)



Theorem  xpsnen 6591 
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 6592 
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 6593 
One times a cardinal number. (Contributed by NM, 27Sep2004.) (Revised
by Mario Carneiro, 29Apr2015.)



Theorem  endisj 6594* 
Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
[Mendelson] p. 255. (Contributed by
NM, 16Apr2004.)



Theorem  xpcomf1o 6595* 
The canonical bijection from to .
(Contributed by Mario Carneiro, 23Apr2014.)



Theorem  xpcomco 6596* 
Composition with the bijection of xpcomf1o 6595 swaps the arguments to a
mapping. (Contributed by Mario Carneiro, 30May2015.)



Theorem  xpcomen 6597 
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 5Jan2004.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  xpcomeng 6598 
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 27Mar2006.)



Theorem  xpsnen2g 6599 
A set is equinumerous to its Cartesian product with a singleton on the
left. (Contributed by Stefan O'Rear, 21Nov2014.)



Theorem  xpassen 6600 
Associative law for equinumerosity of Cartesian product. Proposition
4.22(e) of [Mendelson] p. 254.
(Contributed by NM, 22Jan2004.)
(Revised by Mario Carneiro, 15Nov2014.)

