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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremadd1p1 8601 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
 
Theoremsub1m1 8602 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
 
Theoremcnm2m1cnm3 8603 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 8604 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 8605 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))
 
3.4.6  The Archimedean property
 
Theoremarch 8606* 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 8607* 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 8608* 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)
 
3.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 8609 Extend class notation to include the class of nonnegative integers.
class 0
 
Definitiondf-n0 8610 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})
 
Theoremelnn0 8611 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
 
Theoremnnssnn0 8612 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0
 
Theoremnn0ssre 8613 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ
 
Theoremnn0sscn 8614 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
0 ⊆ ℂ
 
Theoremnn0ex 8615 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V
 
Theoremnnnn0 8616 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
 
Theoremnnnn0i 8617 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0
 
Theoremnn0re 8618 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)
 
Theoremnn0cn 8619 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℂ)
 
Theoremnn0rei 8620 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℝ
 
Theoremnn0cni 8621 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℂ
 
Theoremdfn2 8622 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 8623 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 8624 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 ∈ ℕ0
 
Theorem1nn0 8625 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
1 ∈ ℕ0
 
Theorem2nn0 8626 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
2 ∈ ℕ0
 
Theorem3nn0 8627 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
3 ∈ ℕ0
 
Theorem4nn0 8628 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
4 ∈ ℕ0
 
Theorem5nn0 8629 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
5 ∈ ℕ0
 
Theorem6nn0 8630 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
6 ∈ ℕ0
 
Theorem7nn0 8631 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
7 ∈ ℕ0
 
Theorem8nn0 8632 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
8 ∈ ℕ0
 
Theorem9nn0 8633 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
9 ∈ ℕ0
 
Theoremnn0ge0 8634 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 8635 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 8636 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       0 ≤ 𝑁
 
Theoremnn0le0eq0 8637 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 8638 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 8639 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 8640 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
 
Theorem0mnnnnn0 8641 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 8642 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 8643 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 8644 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 8645 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 8646 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 + 𝑁) ∈ ℕ0
 
Theoremnn0mulcli 8647 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
𝑀 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝑀 · 𝑁) ∈ ℕ0
 
Theoremnn0p1nn 8648 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 8649 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
 
Theoremnnm1nn0 8650 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 8651 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 8652 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))
 
Theoremelnnnn0b 8653 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
 
Theoremelnnnn0c 8654 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
 
Theoremnn0addge1 8655 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 8656 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 8657 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 8658 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 8659 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
𝑁 ∈ ℕ0       𝑁 ≤ (2 · 𝑁)
 
Theoremnn0lele2xi 8660 '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 8661 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 8662 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℕ0)
 
Theoremnn0red 8663 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℝ)
 
Theoremnn0cnd 8664 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℂ)
 
Theoremnn0ge0d 8665 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)       (𝜑 → 0 ≤ 𝐴)
 
Theoremnn0addcld 8666 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 + 𝐵) ∈ ℕ0)
 
Theoremnn0mulcld 8667 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ0)
 
Theoremnn0readdcl 8668 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 8669 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 8670 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 8671 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)
 
3.4.8  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 7473.

 
Syntaxcxnn0 8672 The set of extended nonnegative integers.
class 0*
 
Definitiondf-xnn0 8673 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers *, see df-xr 7473. If we assumed excluded middle, this would be essentially the same as as defined at df-nninf 6738 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
0* = (ℕ0 ∪ {+∞})
 
Theoremelxnn0 8674 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
 
Theoremnn0ssxnn0 8675 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
0 ⊆ ℕ0*
 
Theoremnn0xnn0 8676 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ∈ ℕ0*)
 
Theoremxnn0xr 8677 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)
 
Theorem0xnn0 8678 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
0 ∈ ℕ0*
 
Theorempnf0xnn0 8679 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
+∞ ∈ ℕ0*
 
Theoremnn0nepnf 8680 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ≠ +∞)
 
Theoremnn0xnn0d 8681 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℕ0*)
 
Theoremnn0nepnfd 8682 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ≠ +∞)
 
Theoremxnn0nemnf 8683 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ≠ -∞)
 
Theoremxnn0xrnemnf 8684 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
 
Theoremxnn0nnn0pnf 8685 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 = +∞)
 
3.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 8686 Extend class notation to include the class of integers.
class
 
Definitiondf-z 8687 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 8688 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 
Theoremnnnegz 8689 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
 
Theoremzre 8690 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
 
Theoremzcn 8691 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
 
Theoremzrei 8692 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
𝐴 ∈ ℤ       𝐴 ∈ ℝ
 
Theoremzssre 8693 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 8694 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 8695 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℤ ∈ V
 
Theoremelnnz 8696 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
 
Theorem0z 8697 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0 ∈ ℤ
 
Theorem0zd 8698 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℤ)
 
Theoremelnn0z 8699 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
 
Theoremelznn0nn 8700 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
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