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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfnn3 12101* Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
โ„• = โˆฉ {๐‘ฅ โˆฃ (๐‘ฅ โŠ† โ„ โˆง 1 โˆˆ ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘ฅ (๐‘ฆ + 1) โˆˆ ๐‘ฅ)}
 
Theoremnnred 12102 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„)
 
Theoremnncnd 12103 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„‚)
 
Theorempeano2nnd 12104 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด + 1) โˆˆ โ„•)
 
5.4.2  Principle of mathematical induction
 
Theoremnnind 12105* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 12110 for an example of its use. See nn0ind 12529 for induction on nonnegative integers and uzind 12526, uzind4 12760 for induction on an arbitrary upper set of integers. See indstr 12770 for strong induction. See also nnindALT 12106. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
(๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œƒ))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œ))    &   ๐œ“    &   (๐‘ฆ โˆˆ โ„• โ†’ (๐œ’ โ†’ ๐œƒ))    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œ)
 
TheoremnnindALT 12106* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 12105 has a different hypothesis order. It may be easier to use with the Metamath program Proof Assistant, because "MM-PA> ASSIGN LAST" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> MINIMIZE_WITH nnind / MAYGROW";. (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

(๐‘ฆ โˆˆ โ„• โ†’ (๐œ’ โ†’ ๐œƒ))    &   ๐œ“    &   (๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œƒ))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œ))    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œ)
 
Theoremnnindd 12107* Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(๐‘ฅ = 1 โ†’ (๐œ“ โ†” ๐œ’))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ“ โ†” ๐œƒ))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ“ โ†” ๐œ))    &   (๐‘ฅ = ๐ด โ†’ (๐œ“ โ†” ๐œ‚))    &   (๐œ‘ โ†’ ๐œ’)    &   (((๐œ‘ โˆง ๐‘ฆ โˆˆ โ„•) โˆง ๐œƒ) โ†’ ๐œ)    โ‡’   ((๐œ‘ โˆง ๐ด โˆˆ โ„•) โ†’ ๐œ‚)
 
Theoremnn1m1nn 12108 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
(๐ด โˆˆ โ„• โ†’ (๐ด = 1 โˆจ (๐ด โˆ’ 1) โˆˆ โ„•))
 
Theoremnn1suc 12109* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(๐‘ฅ = 1 โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = ๐ด โ†’ (๐œ‘ โ†” ๐œƒ))    &   ๐œ“    &   (๐‘ฆ โˆˆ โ„• โ†’ ๐œ’)    โ‡’   (๐ด โˆˆ โ„• โ†’ ๐œƒ)
 
Theoremnnaddcl 12110 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด + ๐ต) โˆˆ โ„•)
 
Theoremnnmulcl 12111 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 11049 and ax-mulass 11051. (Revised by Steven Nguyen, 24-Sep-2022.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด ยท ๐ต) โˆˆ โ„•)
 
Theoremnnmulcli 12112 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
๐ด โˆˆ โ„•    &   ๐ต โˆˆ โ„•    โ‡’   (๐ด ยท ๐ต) โˆˆ โ„•
 
Theoremnnmtmip 12113 "Minus times minus is plus, The reason for this we need not discuss." (W. H. Auden, as quoted in M. Guillen "Bridges to Infinity", p. 64, see also Metamath Book, section 1.1.1, p. 5) This statement, formalized to "The product of two negative integers is a positive integer", is proved by the following theorem, therefore it actually need not be discussed anymore. "The reason for this" is that (-๐ด ยท -๐ต) = (๐ด ยท ๐ต) for all complex numbers ๐ด and ๐ต because of mul2neg 11528, ๐ด and ๐ต are complex numbers because of nncn 12095, and (๐ด ยท ๐ต) โˆˆ โ„• because of nnmulcl 12111. This also holds for positive reals, see rpmtmip 12868. Note that the opposites -๐ด and -๐ต of the positive integers ๐ด and ๐ต are negative integers. (Contributed by AV, 23-Dec-2022.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (-๐ด ยท -๐ต) โˆˆ โ„•)
 
Theoremnn2ge 12114* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ โˆƒ๐‘ฅ โˆˆ โ„• (๐ด โ‰ค ๐‘ฅ โˆง ๐ต โ‰ค ๐‘ฅ))
 
Theoremnnge1 12115 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ 1 โ‰ค ๐ด)
 
Theoremnngt1ne1 12116 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
(๐ด โˆˆ โ„• โ†’ (1 < ๐ด โ†” ๐ด โ‰  1))
 
Theoremnnle1eq1 12117 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
(๐ด โˆˆ โ„• โ†’ (๐ด โ‰ค 1 โ†” ๐ด = 1))
 
Theoremnngt0 12118 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
(๐ด โˆˆ โ„• โ†’ 0 < ๐ด)
 
Theoremnnnlt1 12119 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(๐ด โˆˆ โ„• โ†’ ยฌ ๐ด < 1)
 
Theoremnnnle0 12120 A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
(๐ด โˆˆ โ„• โ†’ ยฌ ๐ด โ‰ค 0)
 
Theoremnnne0 12121 A positive integer is nonzero. See nnne0ALT 12125 for a shorter proof using ax-pre-mulgt0 11062. This proof avoids 0lt1 11611, and thus ax-pre-mulgt0 11062, by splitting ax-1ne0 11054 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 11062. (Revised by Steven Nguyen, 30-Jan-2023.)
(๐ด โˆˆ โ„• โ†’ ๐ด โ‰  0)
 
Theoremnnneneg 12122 No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023.)
(๐ด โˆˆ โ„• โ†’ ๐ด โ‰  -๐ด)
 
Theorem0nnn 12123 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 11062. (Revised by Steven Nguyen, 30-Jan-2023.)
ยฌ 0 โˆˆ โ„•
 
Theorem0nnnALT 12124 Alternate proof of 0nnn 12123, which requires ax-pre-mulgt0 11062 but is not based on nnne0 12121 (and which can therefore be used in nnne0ALT 12125). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
ยฌ 0 โˆˆ โ„•
 
Theoremnnne0ALT 12125 Alternate version of nnne0 12121. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
(๐ด โˆˆ โ„• โ†’ ๐ด โ‰  0)
 
Theoremnngt0i 12126 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
๐ด โˆˆ โ„•    โ‡’   0 < ๐ด
 
Theoremnnne0i 12127 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
๐ด โˆˆ โ„•    โ‡’   ๐ด โ‰  0
 
Theoremnndivre 12128 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
((๐ด โˆˆ โ„ โˆง ๐‘ โˆˆ โ„•) โ†’ (๐ด / ๐‘) โˆˆ โ„)
 
Theoremnnrecre 12129 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
(๐‘ โˆˆ โ„• โ†’ (1 / ๐‘) โˆˆ โ„)
 
Theoremnnrecgt0 12130 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
(๐ด โˆˆ โ„• โ†’ 0 < (1 / ๐ด))
 
Theoremnnsub 12131 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด < ๐ต โ†” (๐ต โˆ’ ๐ด) โˆˆ โ„•))
 
Theoremnnsubi 12132 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
๐ด โˆˆ โ„•    &   ๐ต โˆˆ โ„•    โ‡’   (๐ด < ๐ต โ†” (๐ต โˆ’ ๐ด) โˆˆ โ„•)
 
Theoremnndiv 12133* Two ways to express "๐ด divides ๐ต " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„•) โ†’ (โˆƒ๐‘ฅ โˆˆ โ„• (๐ด ยท ๐‘ฅ) = ๐ต โ†” (๐ต / ๐ด) โˆˆ โ„•))
 
Theoremnndivtr 12134 Transitive property of divisibility: if ๐ด divides ๐ต and ๐ต divides ๐ถ, then ๐ด divides ๐ถ. Typically, ๐ถ would be an integer, although the theorem holds for complex ๐ถ. (Contributed by NM, 3-May-2005.)
(((๐ด โˆˆ โ„• โˆง ๐ต โˆˆ โ„• โˆง ๐ถ โˆˆ โ„‚) โˆง ((๐ต / ๐ด) โˆˆ โ„• โˆง (๐ถ / ๐ต) โˆˆ โ„•)) โ†’ (๐ถ / ๐ด) โˆˆ โ„•)
 
Theoremnnge1d 12135 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ 1 โ‰ค ๐ด)
 
Theoremnngt0d 12136 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ 0 < ๐ด)
 
Theoremnnne0d 12137 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ ๐ด โ‰  0)
 
Theoremnnrecred 12138 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (1 / ๐ด) โˆˆ โ„)
 
Theoremnnaddcld 12139 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด + ๐ต) โˆˆ โ„•)
 
Theoremnnmulcld 12140 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด ยท ๐ต) โˆˆ โ„•)
 
Theoremnndivred 12141 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„•)    โ‡’   (๐œ‘ โ†’ (๐ด / ๐ต) โˆˆ โ„)
 
5.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 10992 through df-9 12157), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 10992 and df-1 10993).

With the decimal constructor df-dec 12552, it is possible to easily express larger integers in base 10. See deccl 12566 and the theorems that follow it. See also 4001prm 16952 (4001 is prime) and the proof of bpos 26563. Note that the decimal constructor builds on the definitions in this section.

Note: The number 10 will be represented by its digits using the decimal constructor only, i.e., by 10. Therefore, only decimal digits are needed (as symbols) for the decimal representation of a number.

Integers can also be exhibited as sums of powers of 10 (e.g., the number 103 can be expressed as ((10โ†‘2) + 3)) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as (7โ†‘7) โˆ’ 2. Decimals can be expressed as ratios of integers, as in cos2bnd 16005.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

 
Syntaxc2 12142 Extend class notation to include the number 2.
class 2
 
Syntaxc3 12143 Extend class notation to include the number 3.
class 3
 
Syntaxc4 12144 Extend class notation to include the number 4.
class 4
 
Syntaxc5 12145 Extend class notation to include the number 5.
class 5
 
Syntaxc6 12146 Extend class notation to include the number 6.
class 6
 
Syntaxc7 12147 Extend class notation to include the number 7.
class 7
 
Syntaxc8 12148 Extend class notation to include the number 8.
class 8
 
Syntaxc9 12149 Extend class notation to include the number 9.
class 9
 
Definitiondf-2 12150 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)
 
Definitiondf-3 12151 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)
 
Definitiondf-4 12152 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)
 
Definitiondf-5 12153 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)
 
Definitiondf-6 12154 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)
 
Definitiondf-7 12155 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)
 
Definitiondf-8 12156 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)
 
Definitiondf-9 12157 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)
 
Theorem0ne1 12158 Zero is different from one (the commuted form is Axiom ax-1ne0 11054). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 โ‰  1
 
Theorem1m1e0 12159 One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 โˆ’ 1) = 0
 
Theorem2nn 12160 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 โˆˆ โ„•
 
Theorem2re 12161 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 โˆˆ โ„
 
Theorem2cn 12162 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
2 โˆˆ โ„‚
 
Theorem2cnALT 12163 Alternate proof of 2cn 12162. Shorter but uses more axioms. Similar proofs are possible for 3cn 12168, ... , 9cn 12187. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
2 โˆˆ โ„‚
 
Theorem2ex 12164 The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
2 โˆˆ V
 
Theorem2cnd 12165 The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(๐œ‘ โ†’ 2 โˆˆ โ„‚)
 
Theorem3nn 12166 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 โˆˆ โ„•
 
Theorem3re 12167 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 โˆˆ โ„
 
Theorem3cn 12168 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
3 โˆˆ โ„‚
 
Theorem3ex 12169 The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 โˆˆ V
 
Theorem4nn 12170 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 โˆˆ โ„•
 
Theorem4re 12171 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 โˆˆ โ„
 
Theorem4cn 12172 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
4 โˆˆ โ„‚
 
Theorem5nn 12173 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 โˆˆ โ„•
 
Theorem5re 12174 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 โˆˆ โ„
 
Theorem5cn 12175 The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
5 โˆˆ โ„‚
 
Theorem6nn 12176 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 โˆˆ โ„•
 
Theorem6re 12177 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 โˆˆ โ„
 
Theorem6cn 12178 The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
6 โˆˆ โ„‚
 
Theorem7nn 12179 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 โˆˆ โ„•
 
Theorem7re 12180 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 โˆˆ โ„
 
Theorem7cn 12181 The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
7 โˆˆ โ„‚
 
Theorem8nn 12182 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 โˆˆ โ„•
 
Theorem8re 12183 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 โˆˆ โ„
 
Theorem8cn 12184 The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
8 โˆˆ โ„‚
 
Theorem9nn 12185 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 โˆˆ โ„•
 
Theorem9re 12186 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 โˆˆ โ„
 
Theorem9cn 12187 The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
9 โˆˆ โ„‚
 
Theorem0le0 12188 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 โ‰ค 0
 
Theorem0le2 12189 The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 โ‰ค 2
 
Theorem2pos 12190 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2
 
Theorem2ne0 12191 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 โ‰  0
 
Theorem3pos 12192 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3
 
Theorem3ne0 12193 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 โ‰  0
 
Theorem4pos 12194 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4
 
Theorem4ne0 12195 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 โ‰  0
 
Theorem5pos 12196 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5
 
Theorem6pos 12197 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6
 
Theorem7pos 12198 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7
 
Theorem8pos 12199 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8
 
Theorem9pos 12200 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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