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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsnnen2og 6801 A singleton is never equinumerous with the ordinal number 2. If is a proper class, see snnen2oprc 6802. (Contributed by Jim Kingdon, 1-Sep-2021.)

Theoremsnnen2oprc 6802 A singleton is never equinumerous with the ordinal number 2. If is a set, see snnen2og 6801. (Contributed by Jim Kingdon, 1-Sep-2021.)

Theorem1nen2 6803 One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)

Theoremphplem4dom 6804 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)

Theoremphp5dom 6805 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)

Theoremnndomo 6806 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)

Theoremphpm 6807* 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 6794 through phplem4 6797, nneneq 6799, and this final piece of the proof. (Contributed by NM, 29-May-1998.)

Theoremphpelm 6808 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)

Theoremphplem4on 6809 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)

2.6.31  Finite sets

Theoremfict 6810 A finite set is dominated by . Also see finct 7054. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Theoremfidceq 6811 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.)
DECID

Theoremfidifsnen 6812 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)

Theoremfidifsnid 6813 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 3702 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)

Theoremnnfi 6814 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremenfi 6815 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)

Theoremenfii 6816 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremssfilem 6817* Lemma for ssfiexmid 6818. (Contributed by Jim Kingdon, 3-Feb-2022.)

Theoremssfiexmid 6818* 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.)

Theoreminfiexmid 6819* If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)

Theoremdomfiexmid 6820* If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)

Theoremdif1en 6821 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.)

Theoremdif1enen 6822 Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)

Theoremfiunsnnn 6823 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)

Theoremphp5fin 6824 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)

Theoremfisbth 6825 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)

Theorem0fin 6826 The empty set is finite. (Contributed by FL, 14-Jul-2008.)

Theoremfin0 6827* A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)

Theoremfin0or 6828* A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)

Theoremdiffitest 6829* 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.)

Theoremfindcard 6830* 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.)

Theoremfindcard2 6831* 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.)

Theoremfindcard2s 6832* Variation of findcard2 6831 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremfindcard2d 6833* Deduction version of findcard2 6831. If you also need (which doesn't come for free due to ssfiexmid 6818), use findcard2sd 6834 instead. (Contributed by SO, 16-Jul-2018.)

Theoremfindcard2sd 6834* Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)

Theoremdiffisn 6835 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)

Theoremdiffifi 6836 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)

Theoreminfnfi 6837 An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)

Theoremominf 6838 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 6709. (Contributed by NM, 2-Jun-1998.)

Theoremisinfinf 6839* An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)

Theoremac6sfi 6840* Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Theoremtridc 6841* A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
DECID

Theoremfimax2gtrilemstep 6842* Lemma for fimax2gtri 6843. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)

Theoremfimax2gtri 6843* A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)

Theoremfinexdc 6844* Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
DECID DECID

Theoremdfrex2fin 6845* 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.)
DECID

Theoreminfm 6846* An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)

Theoreminfn0 6847 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)

Theoreminffiexmid 6848* If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.)

Theoremen2eqpr 6849 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremexmidpw 6850 Excluded middle is equivalent to the power set of having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
EXMID

Theoremexmidpweq 6851 Excluded middle is equivalent to the power set of being . (Contributed by Jim Kingdon, 28-Jul-2024.)
EXMID

Theorempw1fin 6852 Excluded middle is equivalent to the power set of being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
EXMID

Theorempw1dc0el 6853 Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
EXMID DECID

Theoremss1o0el1o 6854 Reformulation of ss1o0el1 4158 using instead of . (Contributed by BJ, 9-Aug-2024.)

Theorempw1dc1 6855 If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
EXMID DECID

Theoremfientri3 6856 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)

Theoremnnwetri 6857* A natural number is well-ordered by . More specifically, this order both satisfies and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)

Theoremonunsnss 6858 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)

Theoremunfiexmid 6859* If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)

Theoremunsnfi 6860 Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)

Theoremunsnfidcex 6861 The condition in unsnfi 6860. This is intended to show that unsnfi 6860 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
DECID

Theoremunsnfidcel 6862 The condition in unsnfi 6860. This is intended to show that unsnfi 6860 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
DECID

Theoremunfidisj 6863 The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)

Theoremundifdcss 6864* Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
DECID

Theoremundifdc 6865* Union of complementary parts into whole. This is a case where we can strengthen undifss 3474 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
DECID

Theoremundiffi 6866 Union of complementary parts into whole. This is a case where we can strengthen undifss 3474 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)

Theoremunfiin 6867 The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)

Theoremprfidisj 6868 A pair is finite if it consists of two unequal sets. For the case where , see snfig 6756. For the cases where one or both is a proper class, see prprc1 3667, prprc2 3668, or prprc 3669. (Contributed by Jim Kingdon, 31-May-2022.)

Theoremtpfidisj 6869 A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)

Theoremfiintim 6870* If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as and not being equal, or having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

Theoremxpfi 6871 The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theorem3xpfi 6872 The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)

Theoremfisseneq 6873 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)

Theoremphpeqd 6874 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6807 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Theoremssfirab 6875* A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
DECID

Theoremssfidc 6876* A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
DECID

Theoremsnon0 6877 An ordinal which is a singleton is . (Contributed by Jim Kingdon, 19-Oct-2021.)

Theoremfnfi 6878 A version of fnex 5688 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremfundmfi 6879 The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)

Theoremfundmfibi 6880 A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)

Theoremresfnfinfinss 6881 The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)

Theoremrelcnvfi 6882 If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)

Theoremfunrnfi 6883 The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)

Theoremf1ofi 6884 If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)

Theoremf1dmvrnfibi 6885 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6886. (Contributed by AV, 10-Jan-2020.)

Theoremf1vrnfibi 6886 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6885. (Contributed by AV, 10-Jan-2020.)

Theoremiunfidisj 6887* The finite union of disjoint finite sets is finite. Note that depends on , i.e. can be thought of as . (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
Disj

Theoremf1finf1o 6888 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)

Theoremen1eqsn 6889 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)

Theoremen1eqsnbi 6890 A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)

Theoremsnexxph 6891* A case where the antecedent of snexg 4145 is not needed. The class is from dcextest 4539. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)

Theorempreimaf1ofi 6892 The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)

Theoremfidcenumlemim 6893* Lemma for fidcenum 6897. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
DECID

Theoremfidcenumlemrks 6894* Lemma for fidcenum 6897. Induction step for fidcenumlemrk 6895. (Contributed by Jim Kingdon, 20-Oct-2022.)
DECID

Theoremfidcenumlemrk 6895* Lemma for fidcenum 6897. (Contributed by Jim Kingdon, 20-Oct-2022.)
DECID

Theoremfidcenumlemr 6896* Lemma for fidcenum 6897. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
DECID

Theoremfidcenum 6897* A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
DECID

2.6.32  Schroeder-Bernstein Theorem

Theoremsbthlem1 6898* Lemma for isbth 6908. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlem2 6899* Lemma for isbth 6908. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlemi3 6900* Lemma for isbth 6908. (Contributed by NM, 22-Mar-1998.)
EXMID

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