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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnm2m1cnm3 9201 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝐴 ∈ ℂ → ((𝐴 − 2) − 1) = (𝐴 − 3))
 
Theoremxp1d2m1eqxm1d2 9202 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
(𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2))
 
Theoremdiv4p1lem1div2 9203 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
 
4.4.6  The Archimedean property
 
Theoremarch 9204* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
(𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
 
Theoremnnrecl 9205* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)
 
Theorembndndx 9206* A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
(∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴𝑥) → ∃𝑘 ∈ ℕ 𝐴𝑘)
 
4.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 9207 Extend class notation to include the class of nonnegative integers.
class 0
 
Definitiondf-n0 9208 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})
 
Theoremelnn0 9209 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
 
Theoremnnssnn0 9210 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0
 
Theoremnn0ssre 9211 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ
 
Theoremnn0sscn 9212 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ
 
Theoremnn0ex 9213 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V
 
Theoremnnnn0 9214 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
 
Theoremnnnn0i 9215 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0
 
Theoremnn0re 9216 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)
 
Theoremnn0cn 9217 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℂ)
 
Theoremnn0rei 9218 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℝ
 
Theoremnn0cni 9219 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℂ
 
Theoremdfn2 9220 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
ℕ = (ℕ0 ∖ {0})
 
Theoremelnnne0 9221 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
 
Theorem0nn0 9222 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 ∈ ℕ0
 
Theorem1nn0 9223 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 ∈ ℕ0
 
Theorem2nn0 9224 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 ∈ ℕ0
 
Theorem3nn0 9225 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 ∈ ℕ0
 
Theorem4nn0 9226 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 ∈ ℕ0
 
Theorem5nn0 9227 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 ∈ ℕ0
 
Theorem6nn0 9228 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 ∈ ℕ0
 
Theorem7nn0 9229 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 ∈ ℕ0
 
Theorem8nn0 9230 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 ∈ ℕ0
 
Theorem9nn0 9231 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 ∈ ℕ0
 
Theoremnn0ge0 9232 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
 
Theoremnn0nlt0 9233 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℕ0 → ¬ 𝐴 < 0)
 
Theoremnn0ge0i 9234 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       0 ≤ 𝑁
 
Theoremnn0le0eq0 9235 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
(𝑁 ∈ ℕ0 → (𝑁 ≤ 0 ↔ 𝑁 = 0))
 
Theoremnn0p1gt0 9236 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
 
Theoremnnnn0addcl 9237 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ)
 
Theoremnn0nnaddcl 9238 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
 
Theorem0mnnnnn0 9239 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
(𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0)
 
Theoremun0addcl 9240 If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 + 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 + 𝑁) ∈ 𝑇)
 
Theoremun0mulcl 9241 If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   𝑇 = (𝑆 ∪ {0})    &   ((𝜑 ∧ (𝑀𝑆𝑁𝑆)) → (𝑀 · 𝑁) ∈ 𝑆)       ((𝜑 ∧ (𝑀𝑇𝑁𝑇)) → (𝑀 · 𝑁) ∈ 𝑇)
 
Theoremnn0addcl 9242 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
 
Theoremnn0mulcl 9243 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)
 
Theoremnn0addcli 9244 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 + 𝑁) ∈ ℕ0
 
Theoremnn0mulcli 9245 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 · 𝑁) ∈ ℕ0
 
Theoremnn0p1nn 9246 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
 
Theorempeano2nn0 9247 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
 
Theoremnnm1nn0 9248 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
 
Theoremelnn0nn 9249 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ))
 
Theoremelnnnn0 9250 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))
 
Theoremelnnnn0b 9251 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
 
Theoremelnnnn0c 9252 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
 
Theoremnn0addge1 9253 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁))
 
Theoremnn0addge2 9254 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝑁 + 𝐴))
 
Theoremnn0addge1i 9255 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
𝐴 ∈ ℝ    &   𝑁 ∈ ℕ0       𝐴 ≤ (𝐴 + 𝑁)
 
Theoremnn0addge2i 9256 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
𝐴 ∈ ℝ    &   𝑁 ∈ ℕ0       𝐴 ≤ (𝑁 + 𝐴)
 
Theoremnn0le2xi 9257 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       𝑁 ≤ (2 · 𝑁)
 
Theoremnn0lele2xi 9258 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑁𝑀𝑁 ≤ (2 · 𝑀))
 
Theoremnn0supp 9259 Two ways to write the support of a function on 0. (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐹:𝐼⟶ℕ0 → (𝐹 “ (V ∖ {0})) = (𝐹 “ ℕ))
 
Theoremnnnn0d 9260 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℕ0)
 
Theoremnn0red 9261 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℝ)
 
Theoremnn0cnd 9262 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℂ)
 
Theoremnn0ge0d 9263 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑 → 0 ≤ 𝐴)
 
Theoremnn0addcld 9264 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ0)
 
Theoremnn0mulcld 9265 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ0)
 
Theoremnn0readdcl 9266 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremnn0ge2m1nn 9267 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
 
Theoremnn0ge2m1nn0 9268 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ0)
 
Theoremnn0nndivcl 9269 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℕ0𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ)
 
4.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers *, see df-xr 8027.

 
Syntaxcxnn0 9270 The set of extended nonnegative integers.
class 0*
 
Definitiondf-xnn0 9271 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers *, see df-xr 8027. If we assumed excluded middle, this would be essentially the same as as defined at df-nninf 7150 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
0* = (ℕ0 ∪ {+∞})
 
Theoremelxnn0 9272 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
 
Theoremnn0ssxnn0 9273 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
0 ⊆ ℕ0*
 
Theoremnn0xnn0 9274 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ∈ ℕ0*)
 
Theoremxnn0xr 9275 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)
 
Theorem0xnn0 9276 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
0 ∈ ℕ0*
 
Theorempnf0xnn0 9277 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
+∞ ∈ ℕ0*
 
Theoremnn0nepnf 9278 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ≠ +∞)
 
Theoremnn0xnn0d 9279 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℕ0*)
 
Theoremnn0nepnfd 9280 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ≠ +∞)
 
Theoremxnn0nemnf 9281 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ≠ -∞)
 
Theoremxnn0xrnemnf 9282 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
 
Theoremxnn0nnn0pnf 9283 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞)
 
4.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 9284 Extend class notation to include the class of integers.
class
 
Definitiondf-z 9285 Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
 
Theoremelz 9286 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 
Theoremnnnegz 9287 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
 
Theoremzre 9288 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
 
Theoremzcn 9289 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
 
Theoremzrei 9290 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
𝐴 ∈ ℤ       𝐴 ∈ ℝ
 
Theoremzssre 9291 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 9292 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 9293 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℤ ∈ V
 
Theoremelnnz 9294 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
 
Theorem0z 9295 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0 ∈ ℤ
 
Theorem0zd 9296 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℤ)
 
Theoremelnn0z 9297 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
 
Theoremelznn0nn 9298 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
 
Theoremelznn0 9299 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
 
Theoremelznn 9300 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ0)))
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