P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	avglt2 7901	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ ((A + B) / 2) < B))
Theorem	avgle1 7902	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ B ↔ A ≤ ((A + B) / 2)))
Theorem	avgle2 7903	Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ B ↔ ((A + B) / 2) ≤ B))
Theorem	2timesd 7904	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → (2 · A) = (A + A))
Theorem	times2d 7905	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → (A · 2) = (A + A))
Theorem	halfcld 7906	Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → (A / 2) ∈ ℂ)
Theorem	2halvesd 7907	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℂ)    ⇒   ⊢ (φ → ((A / 2) + (A / 2)) = A)
Theorem	rehalfcld 7908	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A / 2) ∈ ℝ)
Theorem	lt2halvesd 7909	A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 𝐶 ∈ ℝ)    &   ⊢ (φ → A < (𝐶 / 2))    &   ⊢ (φ → B < (𝐶 / 2))    ⇒   ⊢ (φ → (A + B) < 𝐶)
Theorem	rehalfcli 7910	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
⊢ A ∈ ℝ    ⇒   ⊢ (A / 2) ∈ ℝ
Theorem	add1p1 7911	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
Theorem	sub1m1 7912	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))
Theorem	cnm2m1cnm3 7913	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))
3.4.6  The Archimedean property
Theorem	arch 7914*	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)
Theorem	nnrecl 7915*	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)
Theorem	bndndx 7916*	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 ∧ A ≤ x) → ∃𝑘 ∈ ℕ A ≤ 𝑘)
3.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 7917	Extend class notation to include the class of nonnegative integers.
class ℕ0
Definition	df-n0 7918	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 = (ℕ ∪ {0})
Theorem	elnn0 7919	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))
Theorem	nnssnn0 7920	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ ⊆ ℕ0
Theorem	nn0ssre 7921	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 7922	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 7923	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 7924	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ → A ∈ ℕ0)
Theorem	nnnn0i 7925	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 7926	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ0 → A ∈ ℝ)
Theorem	nn0cn 7927	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (A ∈ ℕ0 → A ∈ ℂ)
Theorem	nn0rei 7928	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ A ∈ ℕ0    ⇒   ⊢ A ∈ ℝ
Theorem	nn0cni 7929	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ A ∈ ℕ0    ⇒   ⊢ A ∈ ℂ
Theorem	dfn2 7930	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})
Theorem	elnnne0 7931	The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
Theorem	0nn0 7932	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 7933	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 7934	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 7935	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 7936	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 7937	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 7938	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 7939	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 7940	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 7941	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	10nn0 7942	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 10 ∈ ℕ0
Theorem	nn0ge0 7943	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)
Theorem	nn0nlt0 7944	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)
Theorem	nn0ge0i 7945	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 7946	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))
Theorem	nn0p1gt0 7947	A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
Theorem	nnnn0addcl 7948	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)
Theorem	nn0nnaddcl 7949	A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
Theorem	0mnnnnn0 7950	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)
Theorem	un0addcl 7951	If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (φ → 𝑆 ⊆ ℂ)    &   ⊢ 𝑇 = (𝑆 ∪ {0})    &   ⊢ ((φ ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 + 𝑁) ∈ 𝑆)    ⇒   ⊢ ((φ ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 + 𝑁) ∈ 𝑇)
Theorem	un0mulcl 7952	If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (φ → 𝑆 ⊆ ℂ)    &   ⊢ 𝑇 = (𝑆 ∪ {0})    &   ⊢ ((φ ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ 𝑆)    ⇒   ⊢ ((φ ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 · 𝑁) ∈ 𝑇)
Theorem	nn0addcl 7953	Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
Theorem	nn0mulcl 7954	Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0)
Theorem	nn0addcli 7955	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 7956	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 7957	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)
Theorem	peano2nn0 7958	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 7959	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)
Theorem	elnn0nn 7960	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))
Theorem	elnnnn0 7961	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0))
Theorem	elnnnn0b 7962	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 7963	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 7964	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))
Theorem	nn0addge2 7965	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))
Theorem	nn0addge1i 7966	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)
Theorem	nn0addge2i 7967	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)
Theorem	nn0le2xi 7968	A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝑁 ≤ (2 · 𝑁)
Theorem	nn0lele2xi 7969	'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    ⇒   ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀))
Theorem	nn0supp 7970	Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ))
Theorem	nnnn0d 7971	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℕ0)
Theorem	nn0red 7972	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ0)    ⇒   ⊢ (φ → A ∈ ℝ)
Theorem	nn0cnd 7973	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ0)    ⇒   ⊢ (φ → A ∈ ℂ)
Theorem	nn0ge0d 7974	A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ0)    ⇒   ⊢ (φ → 0 ≤ A)
Theorem	nn0addcld 7975	Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ0)    &   ⊢ (φ → B ∈ ℕ0)    ⇒   ⊢ (φ → (A + B) ∈ ℕ0)
Theorem	nn0mulcld 7976	Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (φ → A ∈ ℕ0)    &   ⊢ (φ → B ∈ ℕ0)    ⇒   ⊢ (φ → (A · B) ∈ ℕ0)
Theorem	nn0readdcl 7977	Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
⊢ ((A ∈ ℕ0 ∧ B ∈ ℕ0) → (A + B) ∈ ℝ)
Theorem	nn0ge2m1nn 7978	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)
Theorem	nn0ge2m1nn0 7979	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)
Theorem	nn0nndivcl 7980	Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ)
3.4.8  Integers (as a subset of complex numbers)
Syntax	cz 7981	Extend class notation to include the class of integers.
class ℤ
Definition	df-z 7982	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)}
Theorem	elz 7983	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Theorem	nnnegz 7984	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
Theorem	zre 7985	An integer is a real. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
Theorem	zcn 7986	An integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
Theorem	zrei 7987	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
⊢ A ∈ ℤ    ⇒   ⊢ A ∈ ℝ
Theorem	zssre 7988	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℝ
Theorem	zsscn 7989	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℂ
Theorem	zex 7990	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℤ ∈ V
Theorem	elnnz 7991	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
Theorem	0z 7992	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ 0 ∈ ℤ
Theorem	0zd 7993	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (φ → 0 ∈ ℤ)
Theorem	elnn0z 7994	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
Theorem	elznn0nn 7995	Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
Theorem	elznn0 7996	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
Theorem	elznn 7997	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)))
Theorem	nnssz 7998	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
⊢ ℕ ⊆ ℤ
Theorem	nn0ssz 7999	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℤ
Theorem	nnz 8000	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)