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	nnnn0 12501	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
Theorem	nnnn0i 12502	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℕ0
Theorem	nn0re 12503	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
Theorem	nn0cn 12504	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
Theorem	nn0rei 12505	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nn0cni 12506	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 12507	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 12508	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 12509	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 0 ∈ ℕ0
Theorem	1nn0 12510	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 1 ∈ ℕ0
Theorem	2nn0 12511	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 2 ∈ ℕ0
Theorem	3nn0 12512	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 3 ∈ ℕ0
Theorem	4nn0 12513	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 4 ∈ ℕ0
Theorem	5nn0 12514	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 5 ∈ ℕ0
Theorem	6nn0 12515	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 6 ∈ ℕ0
Theorem	7nn0 12516	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 7 ∈ ℕ0
Theorem	8nn0 12517	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 8 ∈ ℕ0
Theorem	9nn0 12518	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 9 ∈ ℕ0
Theorem	nn0ge0 12519	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 12520	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 12521	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 0 ≤ 𝑁
Theorem	nn0le0eq0 12522	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 12523	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 12524	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 12525	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 12526	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 12527	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 12528	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 12529	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 12530	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 12531	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 + 𝑁) ∈ ℕ0
Theorem	nn0mulcli 12532	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝑀 · 𝑁) ∈ ℕ0
Theorem	nn0p1nn 12533	A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 12246. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
Theorem	peano2nn0 12534	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
Theorem	nnm1nn0 12535	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 12536	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 12537	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 12538	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁))
Theorem	elnnnn0c 12539	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁))
Theorem	nn0addge1 12540	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 12541	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 12542	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 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 ≤ (𝑁 + 𝐴)
Theorem	nn0sub 12544	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 12545	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 12546	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴)
Theorem	difgtsumgt 12547	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 12548	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 12549	'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 12550	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 12551	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 12552	Version of fcdmnn0supp 12550 avoiding ax-rep 5279 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 12553	Version of fcdmnn0fsupp 12551 avoiding ax-rep 5279 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 12554	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0)
Theorem	nn0red 12555	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nn0cnd 12556	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	nn0ge0d 12557	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 12558	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 12559	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 12560	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 12561	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 12562	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 12563	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 12564	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 12565	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 14321. 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 11274. 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 16976, or for the degree of polynomials, see mdegcl 25992, or for the degree of vertices in graph theory, see vtxdgf 29272.
Syntax	cxnn0 12566	The set of extended nonnegative integers.
class ℕ0*
Definition	df-xnn0 12567	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 11274. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0* = (ℕ0 ∪ {+∞})
Theorem	elxnn0 12568	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 12569	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0 ⊆ ℕ0*
Theorem	nn0xnn0 12570	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
Theorem	xnn0xr 12571	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*)
Theorem	0xnn0 12572	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ 0 ∈ ℕ0*
Theorem	pnf0xnn0 12573	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ +∞ ∈ ℕ0*
Theorem	nn0nepnf 12574	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞)
Theorem	nn0xnn0d 12575	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 12576	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	xnn0nemnf 12577	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞)
Theorem	xnn0xrnemnf 12578	The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
Theorem	xnn0nnn0pnf 12579	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 12580	Extend class notation to include the class of integers.
class ℤ
Definition	df-z 12581	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 12582	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Theorem	nnnegz 12583	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
Theorem	zre 12584	An integer is a real. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
Theorem	zcn 12585	An integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
Theorem	zrei 12586	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
⊢ 𝐴 ∈ ℤ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	zssre 12587	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℝ
Theorem	zsscn 12588	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℂ
Theorem	zex 12589	The set of integers exists. See also zexALT 12600. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℤ ∈ V
Theorem	elnnz 12590	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
Theorem	0z 12591	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ 0 ∈ ℤ
Theorem	0zd 12592	Zero is an integer, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℤ)
Theorem	elnn0z 12593	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
Theorem	elznn0nn 12594	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 12595	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
Theorem	elznn 12596	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 12597	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))
Theorem	elz2 12598*	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 12599	Alternative definition of the integers, based on elz2 12598. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ ℤ = ( − “ (ℕ × ℕ))
Theorem	zexALT 12600	Alternate proof of zex 12589. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℤ ∈ V