P. 114 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0re 11301	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 11302	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 11303	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 11304	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	dfn2 11305	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 11306	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 11307	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 11308	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 11309	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 11310	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 11311	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 11312	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 11313	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 11314	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 11315	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 11316	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	10nn0OLD 11317	Obsolete version of 10nn0 11516 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 10 ∈ ℕ0
Theorem	nn0ge0 11318	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 11319	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 11320	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 11321	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 11322	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 11323	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 11324	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 11325	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 11326	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 11327	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 11328	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 11329	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 11330	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 11331	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 11332	A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 11032. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
Theorem	peano2nn0 11333	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 11334	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 11335	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 11336	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 11337	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 11338	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 11339	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 11340	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 11341	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 11342	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	nn0sub 11343	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0))
Theorem	ltsubnn0 11344	Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐵 < 𝐴 → (𝐴 − 𝐵) ∈ ℕ0))
Theorem	nn0negleid 11345	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴)
Theorem	difgtsumgt 11346	If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵)))
Theorem	nn0le2xi 11347	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 11348	'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	frnnn0supp 11349	Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
Theorem	frnnn0fsupp 11350	A function on ℕ0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.)
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
Theorem	nnnn0d 11351	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0)
Theorem	nn0red 11352	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nn0cnd 11353	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	nn0ge0d 11354	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 11355	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 11356	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 11357	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	nn0n0n1ge2 11358	A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)
Theorem	nn0n0n1ge2b 11359	A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))
Theorem	nn0ge2m1nn 11360	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 11361	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 11362	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)
5.4.8  Extended nonnegative integers The function values of the hash (set size) function are either nonnegative integers or positive infinity, see hashf 13125. 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 ℝ*, see df-xr 10078. The definition of extended nonnegative integers can be used in Ramsey theory, because the Ramsey number is either a nonnegative integer or plus infinity, see ramcl2 15720, or for the degree of polynomials, see mdegcl 23829, or for the degree of vertices in graph theory, see vtxdgf 26367.
Syntax	cxnn0 11363	The set of extended nonnegative integers.
class ℕ0*
Definition	df-xnn0 11364	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 10078. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0* = (ℕ0 ∪ {+∞})
Theorem	elxnn0 11365	An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
Theorem	nn0ssxnn0 11366	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0 ⊆ ℕ0*
Theorem	nn0xnn0 11367	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
Theorem	xnn0xr 11368	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*)
Theorem	0xnn0 11369	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ 0 ∈ ℕ0*
Theorem	pnf0xnn0 11370	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ +∞ ∈ ℕ0*
Theorem	nn0nepnf 11371	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞)
Theorem	nn0xnn0d 11372	A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0*)
Theorem	nn0nepnfd 11373	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	xnn0nemnf 11374	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞)
Theorem	xnn0xrnemnf 11375	The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
Theorem	xnn0nnn0pnf 11376	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 = +∞)
5.4.9  Integers (as a subset of complex numbers)
Syntax	cz 11377	Extend class notation to include the class of integers.
class ℤ
Definition	df-z 11378	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 11379	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Theorem	nnnegz 11380	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
Theorem	zre 11381	An integer is a real. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
Theorem	zcn 11382	An integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
Theorem	zrei 11383	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
⊢ 𝐴 ∈ ℤ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	zssre 11384	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℝ
Theorem	zsscn 11385	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℂ
Theorem	zex 11386	The set of integers exists. See also zexALT 11396. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℤ ∈ V
Theorem	elnnz 11387	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
Theorem	0z 11388	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ 0 ∈ ℤ
Theorem	0zd 11389	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℤ)
Theorem	elnn0z 11390	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
Theorem	elznn0nn 11391	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 11392	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
Theorem	elznn 11393	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	elz2 11394*	Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℤ ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 − 𝑦))
Theorem	dfz2 11395	Alternative definition of the integers, based on elz2 11394. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ ℤ = ( − “ (ℕ × ℕ))
Theorem	zexALT 11396	Alternate proof of zex 11386. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℤ ∈ V
Theorem	nnssz 11397	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
⊢ ℕ ⊆ ℤ
Theorem	nn0ssz 11398	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℤ
Theorem	nnz 11399	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
Theorem	nn0z 11400	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)