Type  Label  Description 
Statement 

Theorem  idssen 6601 
Equality implies equinumerosity. (Contributed by NM, 30Apr1998.)
(Revised by Mario Carneiro, 15Nov2014.)



Theorem  ssdomg 6602 
A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed
by NM, 19Jun1998.) (Revised by Mario Carneiro, 24Jun2015.)



Theorem  ener 6603 
Equinumerosity is an equivalence relation. (Contributed by NM,
19Mar1998.) (Revised by Mario Carneiro, 15Nov2014.)



Theorem  ensymb 6604 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
Mario Carneiro, 26Apr2015.)



Theorem  ensym 6605 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by
NM, 26Oct2003.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  ensymi 6606 
Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed
by NM, 25Sep2004.)



Theorem  ensymd 6607 
Symmetry of equinumerosity. Deduction form of ensym 6605. (Contributed
by David Moews, 1May2017.)



Theorem  entr 6608 
Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92.
(Contributed by NM, 9Jun1998.)



Theorem  domtr 6609 
Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94.
(Contributed by NM, 4Jun1998.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  entri 6610 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr2i 6611 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr3i 6612 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  entr4i 6613 
A chained equinumerosity inference. (Contributed by NM,
25Sep2004.)



Theorem  endomtr 6614 
Transitivity of equinumerosity and dominance. (Contributed by NM,
7Jun1998.)



Theorem  domentr 6615 
Transitivity of dominance and equinumerosity. (Contributed by NM,
7Jun1998.)



Theorem  f1imaeng 6616 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (Contributed by Mario Carneiro, 15May2015.)



Theorem  f1imaen2g 6617 
A onetoone function's image under a subset of its domain is equinumerous
to the subset. (This version of f1imaen 6618 does not need axsetind 4390.)
(Contributed by Mario Carneiro, 16Nov2014.) (Revised by Mario Carneiro,
25Jun2015.)



Theorem  f1imaen 6618 
A onetoone function's image under a subset of its domain is
equinumerous to the subset. (Contributed by NM, 30Sep2004.)



Theorem  en0 6619 
The empty set is equinumerous only to itself. Exercise 1 of
[TakeutiZaring] p. 88.
(Contributed by NM, 27May1998.)



Theorem  ensn1 6620 
A singleton is equinumerous to ordinal one. (Contributed by NM,
4Nov2002.)



Theorem  ensn1g 6621 
A singleton is equinumerous to ordinal one. (Contributed by NM,
23Apr2004.)



Theorem  enpr1g 6622 
has only
one element. (Contributed by FL, 15Feb2010.)



Theorem  en1 6623* 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by NM, 25Jul2004.)



Theorem  en1bg 6624 
A set is equinumerous to ordinal one iff it is a singleton.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  reuen1 6625* 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1 6626 
Two ways to express "exactly one". (Contributed by Stefan O'Rear,
28Oct2014.)



Theorem  euen1b 6627* 
Two ways to express " has a unique element". (Contributed by
Mario Carneiro, 9Apr2015.)



Theorem  en1uniel 6628 
A singleton contains its sole element. (Contributed by Stefan O'Rear,
16Aug2015.)



Theorem  2dom 6629* 
A set that dominates ordinal 2 has at least 2 different members.
(Contributed by NM, 25Jul2004.)



Theorem  fundmen 6630 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 28Jul2004.) (Revised by Mario Carneiro,
15Nov2014.)



Theorem  fundmeng 6631 
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 17Sep2013.)



Theorem  cnven 6632 
A relational set is equinumerous to its converse. (Contributed by Mario
Carneiro, 28Dec2014.)



Theorem  cnvct 6633 
If a set is dominated by , so is its converse. (Contributed by
Thierry Arnoux, 29Dec2016.)



Theorem  fndmeng 6634 
A function is equinumerate to its domain. (Contributed by Paul Chapman,
22Jun2011.)



Theorem  mapsnen 6635 
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 6636 
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 6637 
Two singletons are equinumerous. (Contributed by NM, 9Nov2003.)



Theorem  snfig 6638 
A singleton is finite. For the proper class case, see snprc 3535.
(Contributed by Jim Kingdon, 13Apr2020.)



Theorem  fiprc 6639 
The class of finite sets is a proper class. (Contributed by Jeff
Hankins, 3Oct2008.)



Theorem  unen 6640 
Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
(Contributed by NM, 11Jun1998.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  ssct 6641 
A subset of a set dominated by is dominated by .
(Contributed by Thierry Arnoux, 31Jan2017.)



Theorem  1domsn 6642 
A singleton (whether of a set or a proper class) is dominated by one.
(Contributed by Jim Kingdon, 1Mar2022.)



Theorem  enm 6643* 
A set equinumerous to an inhabited set is inhabited. (Contributed by
Jim Kingdon, 19May2020.)



Theorem  xpsnen 6644 
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 6645 
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 6646 
One times a cardinal number. (Contributed by NM, 27Sep2004.) (Revised
by Mario Carneiro, 29Apr2015.)



Theorem  endisj 6647* 
Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
[Mendelson] p. 255. (Contributed by
NM, 16Apr2004.)



Theorem  xpcomf1o 6648* 
The canonical bijection from to .
(Contributed by Mario Carneiro, 23Apr2014.)



Theorem  xpcomco 6649* 
Composition with the bijection of xpcomf1o 6648 swaps the arguments to a
mapping. (Contributed by Mario Carneiro, 30May2015.)



Theorem  xpcomen 6650 
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 6651 
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 27Mar2006.)



Theorem  xpsnen2g 6652 
A set is equinumerous to its Cartesian product with a singleton on the
left. (Contributed by Stefan O'Rear, 21Nov2014.)



Theorem  xpassen 6653 
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 6654 
Dominance law for Cartesian product. Proposition 10.33(2) of
[TakeutiZaring] p. 92.
(Contributed by NM, 24Jul2004.) (Revised by
Mario Carneiro, 15Nov2014.)



Theorem  xpdom2g 6655 
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by Mario Carneiro, 26Apr2015.)



Theorem  xpdom1g 6656 
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by NM, 25Mar2006.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  xpdom3m 6657* 
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 6658 
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 6659 
Covering implies injection on power sets. (Contributed by Stefan
O'Rear, 6Nov2014.) (Revised by Mario Carneiro, 24Jun2015.)



Theorem  0domg 6660 
Any set dominates the empty set. (Contributed by NM, 26Oct2003.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  dom0 6661 
A set dominated by the empty set is empty. (Contributed by NM,
22Nov2004.)



Theorem  0dom 6662 
Any set dominates the empty set. (Contributed by NM, 26Oct2003.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  enen1 6663 
Equalitylike theorem for equinumerosity. (Contributed by NM,
18Dec2003.)



Theorem  enen2 6664 
Equalitylike theorem for equinumerosity. (Contributed by NM,
18Dec2003.)



Theorem  domen1 6665 
Equalitylike theorem for equinumerosity and dominance. (Contributed by
NM, 8Nov2003.)



Theorem  domen2 6666 
Equalitylike theorem for equinumerosity and dominance. (Contributed by
NM, 8Nov2003.)



2.6.28 Equinumerosity (cont.)


Theorem  xpf1o 6667* 
Construct a bijection on a Cartesian product given bijections on the
factors. (Contributed by Mario Carneiro, 30May2015.)



Theorem  xpen 6668 
Equinumerosity law for Cartesian product. Proposition 4.22(b) of
[Mendelson] p. 254. (Contributed by
NM, 24Jul2004.)



Theorem  mapen 6669 
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 6670 
Orderpreserving property of set exponentiation. (Contributed by Jim
Kingdon, 15Jul2022.)



Theorem  mapxpen 6671 
Equinumerosity law for double set exponentiation. Proposition 10.45 of
[TakeutiZaring] p. 96.
(Contributed by NM, 21Feb2004.) (Revised by
Mario Carneiro, 24Jun2015.)



Theorem  xpmapenlem 6672* 
Lemma for xpmapen 6673. (Contributed by NM, 1May2004.) (Revised
by
Mario Carneiro, 16Nov2014.)



Theorem  xpmapen 6673 
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 6674* 
Equinumerosity of equinumerous subsets of a set. (Contributed by NM,
30Sep2004.) (Revised by Mario Carneiro, 16Nov2014.)



2.6.29 Pigeonhole Principle


Theorem  phplem1 6675 
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 6676 
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 6677 
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 6679. (Contributed by NM,
26May1998.)



Theorem  phplem4 6678 
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 6679 
A natural number is equinumerous to its successor minus any element of
the successor. Version of phplem3 6677 with unnecessary hypotheses
removed. (Contributed by Jim Kingdon, 1Sep2021.)



Theorem  nneneq 6680 
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 6681 
A natural number is not equinumerous to its successor. Corollary
10.21(1) of [TakeutiZaring] p. 90.
(Contributed by NM, 26Jul2004.)



Theorem  snnen2og 6682 
A singleton is never equinumerous with the ordinal
number 2. If
is a proper
class, see snnen2oprc 6683. (Contributed by Jim Kingdon,
1Sep2021.)



Theorem  snnen2oprc 6683 
A singleton is never equinumerous with the ordinal
number 2. If
is a set, see snnen2og 6682. (Contributed by Jim Kingdon,
1Sep2021.)



Theorem  1nen2 6684 
One and two are not equinumerous. (Contributed by Jim Kingdon,
25Jan2022.)



Theorem  phplem4dom 6685 
Dominance of successors implies dominance of the original natural
numbers. (Contributed by Jim Kingdon, 1Sep2021.)



Theorem  php5dom 6686 
A natural number does not dominate its successor. (Contributed by Jim
Kingdon, 1Sep2021.)



Theorem  nndomo 6687 
Cardinal ordering agrees with natural number ordering. Example 3 of
[Enderton] p. 146. (Contributed by NM,
17Jun1998.)



Theorem  phpm 6688* 
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 6675 through phplem4 6678, nneneq 6680, and
this final piece of the proof. (Contributed by NM, 29May1998.)



Theorem  phpelm 6689 
Pigeonhole Principle. A natural number is not equinumerous to an
element of itself. (Contributed by Jim Kingdon, 6Sep2021.)



Theorem  phplem4on 6690 
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 6691 
A finite set is dominated by . Also see finct 6915. (Contributed
by Thierry Arnoux, 27Mar2018.)



Theorem  fidceq 6692 
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 6693 
All decrements of a finite set are equinumerous. (Contributed by Jim
Kingdon, 9Sep2021.)



Theorem  fidifsnid 6694 
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 3613 from
subset to equality when the set is finite. (Contributed by Jim Kingdon,
9Sep2021.)



Theorem  nnfi 6695 
Natural numbers are finite sets. (Contributed by Stefan O'Rear,
21Mar2015.)



Theorem  enfi 6696 
Equinumerous sets have the same finiteness. (Contributed by NM,
22Aug2008.)



Theorem  enfii 6697 
A set equinumerous to a finite set is finite. (Contributed by Mario
Carneiro, 12Mar2015.)



Theorem  ssfilem 6698* 
Lemma for ssfiexmid 6699. (Contributed by Jim Kingdon, 3Feb2022.)



Theorem  ssfiexmid 6699* 
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 6700* 
If the intersection of any finite set and any other set is finite,
excluded middle follows. (Contributed by Jim Kingdon, 5Feb2022.)

