Theorem List for Intuitionistic Logic Explorer - 6801-6900 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | en1uniel 6801 |
A singleton contains its sole element. (Contributed by Stefan O'Rear,
16-Aug-2015.)
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Theorem | 2dom 6802* |
A set that dominates ordinal 2 has at least 2 different members.
(Contributed by NM, 25-Jul-2004.)
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Theorem | fundmen 6803 |
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro,
15-Nov-2014.)
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Theorem | fundmeng 6804 |
A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.
(Contributed by NM, 17-Sep-2013.)
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Theorem | cnven 6805 |
A relational set is equinumerous to its converse. (Contributed by Mario
Carneiro, 28-Dec-2014.)
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Theorem | cnvct 6806 |
If a set is dominated by , so is its converse. (Contributed by
Thierry Arnoux, 29-Dec-2016.)
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Theorem | fndmeng 6807 |
A function is equinumerate to its domain. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | mapsnen 6808 |
Set exponentiation to a singleton exponent is equinumerous to its base.
Exercise 4.43 of [Mendelson] p. 255.
(Contributed by NM, 17-Dec-2003.)
(Revised by Mario Carneiro, 15-Nov-2014.)
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Theorem | map1 6809 |
Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1.
Exercise 4.42(b) of [Mendelson] p.
255. (Contributed by NM,
17-Dec-2003.)
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Theorem | en2sn 6810 |
Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
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Theorem | snfig 6811 |
A singleton is finite. For the proper class case, see snprc 3657.
(Contributed by Jim Kingdon, 13-Apr-2020.)
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Theorem | fiprc 6812 |
The class of finite sets is a proper class. (Contributed by Jeff
Hankins, 3-Oct-2008.)
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Theorem | unen 6813 |
Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.
(Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro,
26-Apr-2015.)
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Theorem | enpr2d 6814 |
A pair with distinct elements is equinumerous to ordinal two.
(Contributed by Rohan Ridenour, 3-Aug-2023.)
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Theorem | ssct 6815 |
A subset of a set dominated by is dominated by .
(Contributed by Thierry Arnoux, 31-Jan-2017.)
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Theorem | 1domsn 6816 |
A singleton (whether of a set or a proper class) is dominated by one.
(Contributed by Jim Kingdon, 1-Mar-2022.)
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Theorem | enm 6817* |
A set equinumerous to an inhabited set is inhabited. (Contributed by
Jim Kingdon, 19-May-2020.)
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Theorem | xpsnen 6818 |
A set is equinumerous to its Cartesian product with a singleton.
Proposition 4.22(c) of [Mendelson] p.
254. (Contributed by NM,
4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
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Theorem | xpsneng 6819 |
A set is equinumerous to its Cartesian product with a singleton.
Proposition 4.22(c) of [Mendelson] p.
254. (Contributed by NM,
22-Oct-2004.)
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Theorem | xp1en 6820 |
One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised
by Mario Carneiro, 29-Apr-2015.)
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Theorem | endisj 6821* |
Any two sets are equinumerous to disjoint sets. Exercise 4.39 of
[Mendelson] p. 255. (Contributed by
NM, 16-Apr-2004.)
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Theorem | xpcomf1o 6822* |
The canonical bijection from   to   .
(Contributed by Mario Carneiro, 23-Apr-2014.)
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Theorem | xpcomco 6823* |
Composition with the bijection of xpcomf1o 6822 swaps the arguments to a
mapping. (Contributed by Mario Carneiro, 30-May-2015.)
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Theorem | xpcomen 6824 |
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 5-Jan-2004.)
(Revised by Mario Carneiro, 15-Nov-2014.)
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Theorem | xpcomeng 6825 |
Commutative law for equinumerosity of Cartesian product. Proposition
4.22(d) of [Mendelson] p. 254.
(Contributed by NM, 27-Mar-2006.)
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Theorem | xpsnen2g 6826 |
A set is equinumerous to its Cartesian product with a singleton on the
left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
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Theorem | xpassen 6827 |
Associative law for equinumerosity of Cartesian product. Proposition
4.22(e) of [Mendelson] p. 254.
(Contributed by NM, 22-Jan-2004.)
(Revised by Mario Carneiro, 15-Nov-2014.)
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Theorem | xpdom2 6828 |
Dominance law for Cartesian product. Proposition 10.33(2) of
[TakeutiZaring] p. 92.
(Contributed by NM, 24-Jul-2004.) (Revised by
Mario Carneiro, 15-Nov-2014.)
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Theorem | xpdom2g 6829 |
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
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Theorem | xpdom1g 6830 |
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro,
26-Apr-2015.)
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Theorem | xpdom3m 6831* |
A set is dominated by its Cartesian product with an inhabited set.
Exercise 6 of [Suppes] p. 98.
(Contributed by Jim Kingdon,
15-Apr-2020.)
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Theorem | xpdom1 6832 |
Dominance law for Cartesian product. Theorem 6L(c) of [Enderton]
p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM,
29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
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Theorem | fopwdom 6833 |
Covering implies injection on power sets. (Contributed by Stefan
O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
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Theorem | 0domg 6834 |
Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.)
(Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | dom0 6835 |
A set dominated by the empty set is empty. (Contributed by NM,
22-Nov-2004.)
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Theorem | 0dom 6836 |
Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.)
(Revised by Mario Carneiro, 26-Apr-2015.)
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Theorem | enen1 6837 |
Equality-like theorem for equinumerosity. (Contributed by NM,
18-Dec-2003.)
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Theorem | enen2 6838 |
Equality-like theorem for equinumerosity. (Contributed by NM,
18-Dec-2003.)
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Theorem | domen1 6839 |
Equality-like theorem for equinumerosity and dominance. (Contributed by
NM, 8-Nov-2003.)
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Theorem | domen2 6840 |
Equality-like theorem for equinumerosity and dominance. (Contributed by
NM, 8-Nov-2003.)
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2.6.29 Equinumerosity (cont.)
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Theorem | xpf1o 6841* |
Construct a bijection on a Cartesian product given bijections on the
factors. (Contributed by Mario Carneiro, 30-May-2015.)
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Theorem | xpen 6842 |
Equinumerosity law for Cartesian product. Proposition 4.22(b) of
[Mendelson] p. 254. (Contributed by
NM, 24-Jul-2004.)
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Theorem | mapen 6843 |
Two set exponentiations are equinumerous when their bases and exponents
are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by
NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
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Theorem | mapdom1g 6844 |
Order-preserving property of set exponentiation. (Contributed by Jim
Kingdon, 15-Jul-2022.)
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Theorem | mapxpen 6845 |
Equinumerosity law for double set exponentiation. Proposition 10.45 of
[TakeutiZaring] p. 96.
(Contributed by NM, 21-Feb-2004.) (Revised by
Mario Carneiro, 24-Jun-2015.)
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Theorem | xpmapenlem 6846* |
Lemma for xpmapen 6847. (Contributed by NM, 1-May-2004.) (Revised
by
Mario Carneiro, 16-Nov-2014.)
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Theorem | xpmapen 6847 |
Equinumerosity law for set exponentiation of a Cartesian product.
Exercise 4.47 of [Mendelson] p. 255.
(Contributed by NM, 23-Feb-2004.)
(Proof shortened by Mario Carneiro, 16-Nov-2014.)
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Theorem | ssenen 6848* |
Equinumerosity of equinumerous subsets of a set. (Contributed by NM,
30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
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2.6.30 Pigeonhole Principle
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Theorem | phplem1 6849 |
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, 25-May-1998.)
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Theorem | phplem2 6850 |
Lemma for Pigeonhole Principle. A natural number is equinumerous to its
successor minus one of its elements. (Contributed by NM, 11-Jun-1998.)
(Revised by Mario Carneiro, 16-Nov-2014.)
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Theorem | phplem3 6851 |
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 6853. (Contributed by NM,
26-May-1998.)
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Theorem | phplem4 6852 |
Lemma for Pigeonhole Principle. Equinumerosity of successors implies
equinumerosity of the original natural numbers. (Contributed by NM,
28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
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Theorem | phplem3g 6853 |
A natural number is equinumerous to its successor minus any element of
the successor. Version of phplem3 6851 with unnecessary hypotheses
removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
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Theorem | nneneq 6854 |
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,
28-May-1998.)
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Theorem | php5 6855 |
A natural number is not equinumerous to its successor. Corollary
10.21(1) of [TakeutiZaring] p. 90.
(Contributed by NM, 26-Jul-2004.)
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Theorem | snnen2og 6856 |
A singleton   is never equinumerous with the ordinal
number 2. If
is a proper
class, see snnen2oprc 6857. (Contributed by Jim Kingdon,
1-Sep-2021.)
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Theorem | snnen2oprc 6857 |
A singleton   is never equinumerous with the ordinal
number 2. If
is a set, see snnen2og 6856. (Contributed by Jim Kingdon,
1-Sep-2021.)
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Theorem | 1nen2 6858 |
One and two are not equinumerous. (Contributed by Jim Kingdon,
25-Jan-2022.)
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Theorem | phplem4dom 6859 |
Dominance of successors implies dominance of the original natural
numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
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Theorem | php5dom 6860 |
A natural number does not dominate its successor. (Contributed by Jim
Kingdon, 1-Sep-2021.)
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Theorem | nndomo 6861 |
Cardinal ordering agrees with natural number ordering. Example 3 of
[Enderton] p. 146. (Contributed by NM,
17-Jun-1998.)
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Theorem | phpm 6862* |
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
so-called because you can't put n +
1 pigeons into n holes (if each hole holds only one pigeon). The
proof consists of lemmas phplem1 6849 through phplem4 6852, nneneq 6854, and
this final piece of the proof. (Contributed by NM, 29-May-1998.)
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Theorem | phpelm 6863 |
Pigeonhole Principle. A natural number is not equinumerous to an
element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
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Theorem | phplem4on 6864 |
Equinumerosity of successors of an ordinal and a natural number implies
equinumerosity of the originals. (Contributed by Jim Kingdon,
5-Sep-2021.)
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2.6.31 Finite sets
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Theorem | fict 6865 |
A finite set is dominated by . Also see finct 7112. (Contributed
by Thierry Arnoux, 27-Mar-2018.)
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Theorem | fidceq 6866 |
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, 5-Sep-2021.)
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Theorem | fidifsnen 6867 |
All decrements of a finite set are equinumerous. (Contributed by Jim
Kingdon, 9-Sep-2021.)
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Theorem | fidifsnid 6868 |
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 3738 from
subset to equality when the set is finite. (Contributed by Jim Kingdon,
9-Sep-2021.)
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Theorem | nnfi 6869 |
Natural numbers are finite sets. (Contributed by Stefan O'Rear,
21-Mar-2015.)
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Theorem | enfi 6870 |
Equinumerous sets have the same finiteness. (Contributed by NM,
22-Aug-2008.)
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Theorem | enfii 6871 |
A set equinumerous to a finite set is finite. (Contributed by Mario
Carneiro, 12-Mar-2015.)
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Theorem | ssfilem 6872* |
Lemma for ssfiexmid 6873. (Contributed by Jim Kingdon, 3-Feb-2022.)
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Theorem | ssfiexmid 6873* |
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, 19-May-2020.)
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Theorem | infiexmid 6874* |
If the intersection of any finite set and any other set is finite,
excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
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Theorem | domfiexmid 6875* |
If any set dominated by a finite set is finite, excluded middle follows.
(Contributed by Jim Kingdon, 3-Feb-2022.)
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Theorem | dif1en 6876 |
If a set is
equinumerous to the successor of a natural number
, then with an element removed is
equinumerous to .
(Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear,
16-Aug-2015.)
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Theorem | dif1enen 6877 |
Subtracting one element from each of two equinumerous finite sets.
(Contributed by Jim Kingdon, 5-Jun-2022.)
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Theorem | fiunsnnn 6878 |
Adding one element to a finite set which is equinumerous to a natural
number. (Contributed by Jim Kingdon, 13-Sep-2021.)
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Theorem | php5fin 6879 |
A finite set is not equinumerous to a set which adds one element.
(Contributed by Jim Kingdon, 13-Sep-2021.)
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Theorem | fisbth 6880 |
Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim
Kingdon, 12-Sep-2021.)
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Theorem | 0fin 6881 |
The empty set is finite. (Contributed by FL, 14-Jul-2008.)
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Theorem | fin0 6882* |
A nonempty finite set has at least one element. (Contributed by Jim
Kingdon, 10-Sep-2021.)
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Theorem | fin0or 6883* |
A finite set is either empty or inhabited. (Contributed by Jim Kingdon,
30-Sep-2021.)
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Theorem | diffitest 6884* |
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,
8-Sep-2021.)
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Theorem | findcard 6885* |
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, 2-Sep-2009.)
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Theorem | findcard2 6886* |
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, 8-Jul-2010.)
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Theorem | findcard2s 6887* |
Variation of findcard2 6886 requiring that the element added in the
induction step not be a member of the original set. (Contributed by
Paul Chapman, 30-Nov-2012.)
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Theorem | findcard2d 6888* |
Deduction version of findcard2 6886. If you also need
(which
doesn't come for free due to ssfiexmid 6873), use findcard2sd 6889 instead.
(Contributed by SO, 16-Jul-2018.)
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Theorem | findcard2sd 6889* |
Deduction form of finite set induction . (Contributed by Jim Kingdon,
14-Sep-2021.)
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Theorem | diffisn 6890 |
Subtracting a singleton from a finite set produces a finite set.
(Contributed by Jim Kingdon, 11-Sep-2021.)
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Theorem | diffifi 6891 |
Subtracting one finite set from another produces a finite set.
(Contributed by Jim Kingdon, 8-Sep-2021.)
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Theorem | infnfi 6892 |
An infinite set is not finite. (Contributed by Jim Kingdon,
20-Feb-2022.)
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Theorem | ominf 6893 |
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 6764. (Contributed by NM,
2-Jun-1998.)
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Theorem | isinfinf 6894* |
An infinite set contains subsets of arbitrarily large finite
cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
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Theorem | ac6sfi 6895* |
Existence of a choice function for finite sets. (Contributed by Jeff
Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro,
29-Jan-2014.)
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Theorem | tridc 6896* |
A trichotomous order is decidable. (Contributed by Jim Kingdon,
5-Sep-2022.)
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Theorem | fimax2gtrilemstep 6897* |
Lemma for fimax2gtri 6898. The induction step. (Contributed by Jim
Kingdon, 5-Sep-2022.)
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Theorem | fimax2gtri 6898* |
A finite set has a maximum under a trichotomous order. (Contributed
by Jim Kingdon, 5-Sep-2022.)
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Theorem | finexdc 6899* |
Decidability of existence, over a finite set and defined by a decidable
proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
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Theorem | dfrex2fin 6900* |
Relationship between universal and existential quantifiers over a finite
set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does
not mention the decidability condition explicitly, it does say
"only
finitely many x to check" which means there must be some way of
checking
each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
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