Type  Label  Description 
Statement 

Theorem  en1bg 6701 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  reuen1 6702* 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1 6703 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1b 6704* 
Two ways to express " has a unique element". (Contributed by
Mario Carneiro, 9Apr2015.)



Theorem  en1uniel 6705 
A singleton contains its sole element. (Contributed by Stefan O'Rear,
16Aug2015.)



Theorem  2dom 6706* 
A set that dominates ordinal 2 has at least 2 different members.
(Contributed by NM, 25Jul2004.)



Theorem  fundmen 6707 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 28Jul2004.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  fundmeng 6708 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 17Sep2013.)



Theorem  cnven 6709 
A relational set is equinumerous to its converse. (Contributed by Mario
Carneiro, 28Dec2014.)



Theorem  cnvct 6710 
If a set is dominated by , so is its converse. (Contributed by
Thierry Arnoux, 29Dec2016.)



Theorem  fndmeng 6711 
A function is equinumerate to its domain. (Contributed by Paul Chapman,
22Jun2011.)



Theorem  mapsnen 6712 
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 6713 
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 6714 
Two singletons are equinumerous. (Contributed by NM, 9Nov2003.)



Theorem  snfig 6715 
A singleton is finite. For the proper class case, see snprc 3595.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  fiprc 6716 
The class of finite sets is a proper class. (Contributed by Jeff
Hankins, 3Oct2008.)



Theorem  unen 6717 
Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
(Contributed by NM, 11Jun1998.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  enpr2d 6718 
A pair with distinct elements is equinumerous to ordinal two.
(Contributed by Rohan Ridenour, 3Aug2023.)



Theorem  ssct 6719 
A subset of a set dominated by is dominated by .
(Contributed by Thierry Arnoux, 31Jan2017.)



Theorem  1domsn 6720 
A singleton (whether of a set or a proper class) is dominated by one.
(Contributed by Jim Kingdon, 1Mar2022.)



Theorem  enm 6721* 
A set equinumerous to an inhabited set is inhabited. (Contributed by
Jim Kingdon, 19May2020.)



Theorem  xpsnen 6722 
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 6723 
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 6724 
One times a cardinal number. (Contributed by NM, 27Sep2004.) (Revised
by Mario Carneiro, 29Apr2015.)



Theorem  endisj 6725* 
Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
[Mendelson] p. 255. (Contributed by
NM, 16Apr2004.)



Theorem  xpcomf1o 6726* 
The canonical bijection from to .
(Contributed by Mario Carneiro, 23Apr2014.)



Theorem  xpcomco 6727* 
Composition with the bijection of xpcomf1o 6726 swaps the arguments to a
mapping. (Contributed by Mario Carneiro, 30May2015.)



Theorem  xpcomen 6728 
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 6729 
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 27Mar2006.)



Theorem  xpsnen2g 6730 
A set is equinumerous to its Cartesian product with a singleton on the
left. (Contributed by Stefan O'Rear, 21Nov2014.)



Theorem  xpassen 6731 
Associative law for equinumerosity of Cartesian product. Proposition
4.22(e) of [Mendelson] p. 254.
(Contributed by NM, 22Jan2004.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  xpdom2 6732 
Dominance law for Cartesian product. Proposition 10.33(2) of
[TakeutiZaring] p. 92.
(Contributed by NM, 24Jul2004.) (Revised by
Mario Carneiro, 15Nov2014.)



Theorem  xpdom2g 6733 
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by Mario Carneiro, 26Apr2015.)



Theorem  xpdom1g 6734 
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by NM, 25Mar2006.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  xpdom3m 6735* 
A set is dominated by its Cartesian product with an inhabited set.
Exercise 6 of [Suppes] p. 98.
(Contributed by Jim Kingdon,
15Apr2020.)



Theorem  xpdom1 6736 
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by NM, 28Sep2004.) (Revised by NM,
29Mar2006.) (Revised by Mario Carneiro, 7May2015.)



Theorem  fopwdom 6737 
Covering implies injection on power sets. (Contributed by Stefan
O'Rear, 6Nov2014.) (Revised by Mario Carneiro, 24Jun2015.)



Theorem  0domg 6738 
Any set dominates the empty set. (Contributed by NM, 26Oct2003.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  dom0 6739 
A set dominated by the empty set is empty. (Contributed by NM,
22Nov2004.)



Theorem  0dom 6740 
Any set dominates the empty set. (Contributed by NM, 26Oct2003.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  enen1 6741 
Equalitylike theorem for equinumerosity. (Contributed by NM,
18Dec2003.)



Theorem  enen2 6742 
Equalitylike theorem for equinumerosity. (Contributed by NM,
18Dec2003.)



Theorem  domen1 6743 
Equalitylike theorem for equinumerosity and dominance. (Contributed by
NM, 8Nov2003.)



Theorem  domen2 6744 
Equalitylike theorem for equinumerosity and dominance. (Contributed by
NM, 8Nov2003.)



2.6.28 Equinumerosity (cont.)


Theorem  xpf1o 6745* 
Construct a bijection on a Cartesian product given bijections on the
factors. (Contributed by Mario Carneiro, 30May2015.)



Theorem  xpen 6746 
Equinumerosity law for Cartesian product. Proposition 4.22(b) of
[Mendelson] p. 254. (Contributed by
NM, 24Jul2004.)



Theorem  mapen 6747 
Two set exponentiations are equinumerous when their bases and exponents
are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by
NM, 16Dec2003.) (Proof shortened by Mario Carneiro, 26Apr2015.)



Theorem  mapdom1g 6748 
Orderpreserving property of set exponentiation. (Contributed by Jim
Kingdon, 15Jul2022.)



Theorem  mapxpen 6749 
Equinumerosity law for double set exponentiation. Proposition 10.45 of
[TakeutiZaring] p. 96.
(Contributed by NM, 21Feb2004.) (Revised by
Mario Carneiro, 24Jun2015.)



Theorem  xpmapenlem 6750* 
Lemma for xpmapen 6751. (Contributed by NM, 1May2004.) (Revised
by
Mario Carneiro, 16Nov2014.)



Theorem  xpmapen 6751 
Equinumerosity law for set exponentiation of a Cartesian product.
Exercise 4.47 of [Mendelson] p. 255.
(Contributed by NM, 23Feb2004.)
(Proof shortened by Mario Carneiro, 16Nov2014.)



Theorem  ssenen 6752* 
Equinumerosity of equinumerous subsets of a set. (Contributed by NM,
30Sep2004.) (Revised by Mario Carneiro, 16Nov2014.)



2.6.29 Pigeonhole Principle


Theorem  phplem1 6753 
Lemma for Pigeonhole Principle. If we join a natural number to itself
minus an element, we end up with its successor minus the same element.
(Contributed by NM, 25May1998.)



Theorem  phplem2 6754 
Lemma for Pigeonhole Principle. A natural number is equinumerous to its
successor minus one of its elements. (Contributed by NM, 11Jun1998.)
(Revised by Mario Carneiro, 16Nov2014.)



Theorem  phplem3 6755 
Lemma for Pigeonhole Principle. A natural number is equinumerous to its
successor minus any element of the successor. For a version without the
redundant hypotheses, see phplem3g 6757. (Contributed by NM,
26May1998.)



Theorem  phplem4 6756 
Lemma for Pigeonhole Principle. Equinumerosity of successors implies
equinumerosity of the original natural numbers. (Contributed by NM,
28May1998.) (Revised by Mario Carneiro, 24Jun2015.)



Theorem  phplem3g 6757 
A natural number is equinumerous to its successor minus any element of
the successor. Version of phplem3 6755 with unnecessary hypotheses
removed. (Contributed by Jim Kingdon, 1Sep2021.)



Theorem  nneneq 6758 
Two equinumerous natural numbers are equal. Proposition 10.20 of
[TakeutiZaring] p. 90 and its
converse. Also compare Corollary 6E of
[Enderton] p. 136. (Contributed by NM,
28May1998.)



Theorem  php5 6759 
A natural number is not equinumerous to its successor. Corollary
10.21(1) of [TakeutiZaring] p. 90.
(Contributed by NM, 26Jul2004.)



Theorem  snnen2og 6760 
A singleton is never equinumerous with the ordinal
number 2. If
is a proper
class, see snnen2oprc 6761. (Contributed by Jim Kingdon,
1Sep2021.)



Theorem  snnen2oprc 6761 
A singleton is never equinumerous with the ordinal
number 2. If
is a set, see snnen2og 6760. (Contributed by Jim Kingdon,
1Sep2021.)



Theorem  1nen2 6762 
One and two are not equinumerous. (Contributed by Jim Kingdon,
25Jan2022.)



Theorem  phplem4dom 6763 
Dominance of successors implies dominance of the original natural
numbers. (Contributed by Jim Kingdon, 1Sep2021.)



Theorem  php5dom 6764 
A natural number does not dominate its successor. (Contributed by Jim
Kingdon, 1Sep2021.)



Theorem  nndomo 6765 
Cardinal ordering agrees with natural number ordering. Example 3 of
[Enderton] p. 146. (Contributed by NM,
17Jun1998.)



Theorem  phpm 6766* 
Pigeonhole Principle. A natural number is not equinumerous to a proper
subset of itself. By "proper subset" here we mean that there
is an
element which is in the natural number and not in the subset, or in
symbols (which is stronger than not being equal
in the absence of excluded middle). Theorem (Pigeonhole Principle) of
[Enderton] p. 134. The theorem is
socalled because you can't put n +
1 pigeons into n holes (if each hole holds only one pigeon). The
proof consists of lemmas phplem1 6753 through phplem4 6756, nneneq 6758, and
this final piece of the proof. (Contributed by NM, 29May1998.)



Theorem  phpelm 6767 
Pigeonhole Principle. A natural number is not equinumerous to an
element of itself. (Contributed by Jim Kingdon, 6Sep2021.)



Theorem  phplem4on 6768 
Equinumerosity of successors of an ordinal and a natural number implies
equinumerosity of the originals. (Contributed by Jim Kingdon,
5Sep2021.)



2.6.30 Finite sets


Theorem  fict 6769 
A finite set is dominated by . Also see finct 7008. (Contributed
by Thierry Arnoux, 27Mar2018.)



Theorem  fidceq 6770 
Equality of members of a finite set is decidable. This may be
counterintuitive: cannot any two sets be elements of a finite set?
Well, to show, for example, that is finite would require
showing it is equinumerous to or to but to show that you'd
need to know
or , respectively.
(Contributed by
Jim Kingdon, 5Sep2021.)

DECID 

Theorem  fidifsnen 6771 
All decrements of a finite set are equinumerous. (Contributed by Jim
Kingdon, 9Sep2021.)



Theorem  fidifsnid 6772 
If we remove a single element from a finite set then put it back in, we
end up with the original finite set. This strengthens difsnss 3673 from
subset to equality when the set is finite. (Contributed by Jim Kingdon,
9Sep2021.)



Theorem  nnfi 6773 
Natural numbers are finite sets. (Contributed by Stefan O'Rear,
21Mar2015.)



Theorem  enfi 6774 
Equinumerous sets have the same finiteness. (Contributed by NM,
22Aug2008.)



Theorem  enfii 6775 
A set equinumerous to a finite set is finite. (Contributed by Mario
Carneiro, 12Mar2015.)



Theorem  ssfilem 6776* 
Lemma for ssfiexmid 6777. (Contributed by Jim Kingdon, 3Feb2022.)



Theorem  ssfiexmid 6777* 
If any subset of a finite set is finite, excluded middle follows. One
direction of Theorem 2.1 of [Bauer], p.
485. (Contributed by Jim
Kingdon, 19May2020.)



Theorem  infiexmid 6778* 
If the intersection of any finite set and any other set is finite,
excluded middle follows. (Contributed by Jim Kingdon, 5Feb2022.)



Theorem  domfiexmid 6779* 
If any set dominated by a finite set is finite, excluded middle follows.
(Contributed by Jim Kingdon, 3Feb2022.)



Theorem  dif1en 6780 
If a set is
equinumerous to the successor of a natural number
, then with an element removed is
equinumerous to .
(Contributed by Jeff Madsen, 2Sep2009.) (Revised by Stefan O'Rear,
16Aug2015.)



Theorem  dif1enen 6781 
Subtracting one element from each of two equinumerous finite sets.
(Contributed by Jim Kingdon, 5Jun2022.)



Theorem  fiunsnnn 6782 
Adding one element to a finite set which is equinumerous to a natural
number. (Contributed by Jim Kingdon, 13Sep2021.)



Theorem  php5fin 6783 
A finite set is not equinumerous to a set which adds one element.
(Contributed by Jim Kingdon, 13Sep2021.)



Theorem  fisbth 6784 
SchroederBernstein Theorem for finite sets. (Contributed by Jim
Kingdon, 12Sep2021.)



Theorem  0fin 6785 
The empty set is finite. (Contributed by FL, 14Jul2008.)



Theorem  fin0 6786* 
A nonempty finite set has at least one element. (Contributed by Jim
Kingdon, 10Sep2021.)



Theorem  fin0or 6787* 
A finite set is either empty or inhabited. (Contributed by Jim Kingdon,
30Sep2021.)



Theorem  diffitest 6788* 
If subtracting any set from a finite set gives a finite set, any
proposition of the form is
decidable. This is not a proof of
full excluded middle, but it is close enough to show we won't be able to
prove . (Contributed by Jim
Kingdon,
8Sep2021.)



Theorem  findcard 6789* 
Schema for induction on the cardinality of a finite set. The inductive
hypothesis is that the result is true on the given set with any one
element removed. The result is then proven to be true for all finite
sets. (Contributed by Jeff Madsen, 2Sep2009.)



Theorem  findcard2 6790* 
Schema for induction on the cardinality of a finite set. The inductive
step shows that the result is true if one more element is added to the
set. The result is then proven to be true for all finite sets.
(Contributed by Jeff Madsen, 8Jul2010.)



Theorem  findcard2s 6791* 
Variation of findcard2 6790 requiring that the element added in the
induction step not be a member of the original set. (Contributed by
Paul Chapman, 30Nov2012.)



Theorem  findcard2d 6792* 
Deduction version of findcard2 6790. If you also need
(which
doesn't come for free due to ssfiexmid 6777), use findcard2sd 6793 instead.
(Contributed by SO, 16Jul2018.)



Theorem  findcard2sd 6793* 
Deduction form of finite set induction . (Contributed by Jim Kingdon,
14Sep2021.)



Theorem  diffisn 6794 
Subtracting a singleton from a finite set produces a finite set.
(Contributed by Jim Kingdon, 11Sep2021.)



Theorem  diffifi 6795 
Subtracting one finite set from another produces a finite set.
(Contributed by Jim Kingdon, 8Sep2021.)



Theorem  infnfi 6796 
An infinite set is not finite. (Contributed by Jim Kingdon,
20Feb2022.)



Theorem  ominf 6797 
The set of natural numbers is not finite. Although we supply this theorem
because we can, the more natural way to express " is infinite" is
which is an instance
of domrefg 6668. (Contributed by NM,
2Jun1998.)



Theorem  isinfinf 6798* 
An infinite set contains subsets of arbitrarily large finite
cardinality. (Contributed by Jim Kingdon, 15Jun2022.)



Theorem  ac6sfi 6799* 
Existence of a choice function for finite sets. (Contributed by Jeff
Hankins, 26Jun2009.) (Proof shortened by Mario Carneiro,
29Jan2014.)



Theorem  tridc 6800* 
A trichotomous order is decidable. (Contributed by Jim Kingdon,
5Sep2022.)

DECID 