P. 126 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0ssre 12501	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ ℕ0 ⊆ ℝ
Theorem	nn0sscn 12502	Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
⊢ ℕ0 ⊆ ℂ
Theorem	nn0ex 12503	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
⊢ ℕ0 ∈ V
Theorem	nnnn0 12504	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 12505	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 12506	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 12507	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 12508	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 12509	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	dfn2 12510	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 12511	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 12512	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 12513	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 12514	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 12515	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 12516	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 12517	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 12518	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 12519	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 12520	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 12521	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	nn0ge0 12522	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 12523	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 12524	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 12525	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 12526	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 12527	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 12528	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 12529	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 12530	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 12531	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 12532	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 12533	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 12534	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 12535	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 12536	A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 12249. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
Theorem	peano2nn0 12537	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 12538	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 12539	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 12540	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 12541	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 12542	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 12543	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 12544	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 12545	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 12546	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 12547	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 12548	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 12549	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴)
Theorem	difgtsumgt 12550	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 12551	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 12552	'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	fcdmnn0supp 12553	Two ways to write the support of a function into 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	fcdmnn0fsupp 12554	A function into 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	fcdmnn0suppg 12555	Version of fcdmnn0supp 12553 avoiding ax-rep 5281 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
Theorem	fcdmnn0fsuppg 12556	Version of fcdmnn0fsupp 12554 avoiding ax-rep 5281 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
Theorem	nnnn0d 12557	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0)
Theorem	nn0red 12558	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nn0cnd 12559	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	nn0ge0d 12560	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 12561	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 12562	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 12563	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 12564	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 12565	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 12566	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 12567	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 12568	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 14324. 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 11277. 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 16979, or for the degree of polynomials, see mdegcl 26018, or for the degree of vertices in graph theory, see vtxdgf 29324.
Syntax	cxnn0 12569	The set of extended nonnegative integers.
class ℕ0*
Definition	df-xnn0 12570	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 11277. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0* = (ℕ0 ∪ {+∞})
Theorem	elxnn0 12571	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 12572	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0 ⊆ ℕ0*
Theorem	nn0xnn0 12573	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
Theorem	xnn0xr 12574	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*)
Theorem	0xnn0 12575	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ 0 ∈ ℕ0*
Theorem	pnf0xnn0 12576	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ +∞ ∈ ℕ0*
Theorem	nn0nepnf 12577	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞)
Theorem	nn0xnn0d 12578	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 12579	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	xnn0nemnf 12580	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞)
Theorem	xnn0xrnemnf 12581	The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
Theorem	xnn0nnn0pnf 12582	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 12583	Extend class notation to include the class of integers.
class ℤ
Definition	df-z 12584	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 12585	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Theorem	nnnegz 12586	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
Theorem	zre 12587	An integer is a real. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
Theorem	zcn 12588	An integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
Theorem	zrei 12589	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
⊢ 𝐴 ∈ ℤ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	zssre 12590	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℝ
Theorem	zsscn 12591	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℂ
Theorem	zex 12592	The set of integers exists. See also zexALT 12603. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℤ ∈ V
Theorem	elnnz 12593	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
Theorem	0z 12594	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ 0 ∈ ℤ
Theorem	0zd 12595	Zero is an integer, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℤ)
Theorem	elnn0z 12596	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
Theorem	elznn0nn 12597	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 12598	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
Theorem	elznn 12599	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	zle0orge1 12600	There is no integer in the open unit interval, i.e., an integer is either less than or equal to 0 or greater than or equal to 1. (Contributed by AV, 4-Jun-2023.)
⊢ (𝑍 ∈ ℤ → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍))