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	fcdmnn0suppg 12501	Version of fcdmnn0supp 12499 avoiding ax-rep 5234 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 12502	Version of fcdmnn0fsupp 12500 avoiding ax-rep 5234 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 12503	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0)
Theorem	nn0red 12504	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nn0cnd 12505	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	nn0ge0d 12506	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 12507	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 12508	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 12509	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 12510	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 12511	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 12512	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 12513	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 12514	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 14303. 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 11212. 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 16987, or for the degree of polynomials, see mdegcl 25974, or for the degree of vertices in graph theory, see vtxdgf 29399.
Syntax	cxnn0 12515	The set of extended nonnegative integers.
class ℕ0*
Definition	df-xnn0 12516	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 11212. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0* = (ℕ0 ∪ {+∞})
Theorem	elxnn0 12517	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 12518	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
⊢ ℕ0 ⊆ ℕ0*
Theorem	nn0xnn0 12519	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
Theorem	xnn0xr 12520	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*)
Theorem	0xnn0 12521	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ 0 ∈ ℕ0*
Theorem	pnf0xnn0 12522	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
⊢ +∞ ∈ ℕ0*
Theorem	nn0nepnf 12523	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞)
Theorem	nn0xnn0d 12524	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 12525	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	xnn0nemnf 12526	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞)
Theorem	xnn0xrnemnf 12527	The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
Theorem	xnn0nnn0pnf 12528	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 12529	Extend class notation to include the class of integers.
class ℤ
Definition	df-z 12530	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 12531	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Theorem	nnnegz 12532	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
Theorem	zre 12533	An integer is a real. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
Theorem	zcn 12534	An integer is a complex number. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
Theorem	zrei 12535	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
⊢ 𝐴 ∈ ℤ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	zssre 12536	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℝ
Theorem	zsscn 12537	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℤ ⊆ ℂ
Theorem	zex 12538	The set of integers exists. See also zexALT 12549. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℤ ∈ V
Theorem	elnnz 12539	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
Theorem	0z 12540	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
⊢ 0 ∈ ℤ
Theorem	0zd 12541	Zero is an integer, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℤ)
Theorem	elnn0z 12542	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
Theorem	elznn0nn 12543	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 12544	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
Theorem	elznn 12545	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 12546	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 12547*	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 12548	Alternative definition of the integers, based on elz2 12547. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ ℤ = ( − “ (ℕ × ℕ))
Theorem	zexALT 12549	Alternate proof of zex 12538. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℤ ∈ V
Theorem	nnz 12550	A positive integer is an integer. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
Theorem	nnssz 12551	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
⊢ ℕ ⊆ ℤ
Theorem	nn0ssz 12552	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
⊢ ℕ0 ⊆ ℤ
Theorem	nnzOLD 12553	Obsolete version of nnz 12550 as of 1-Feb-2025. (Contributed by NM, 9-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
Theorem	nn0z 12554	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
Theorem	nn0zd 12555	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	nnzd 12556	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	nnzi 12557	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℤ
Theorem	nn0zi 12558	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝑁 ∈ ℤ
Theorem	elnnz1 12559	Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁))
Theorem	znnnlt1 12560	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1))
Theorem	nnzrab 12561	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥}
Theorem	nn0zrab 12562	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥}
Theorem	1z 12563	One is an integer. (Contributed by NM, 10-May-2004.)
⊢ 1 ∈ ℤ
Theorem	1zzd 12564	One is an integer, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℤ)
Theorem	2z 12565	2 is an integer. (Contributed by NM, 10-May-2004.)
⊢ 2 ∈ ℤ
Theorem	3z 12566	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ ℤ
Theorem	4z 12567	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
⊢ 4 ∈ ℤ
Theorem	znegcl 12568	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
Theorem	neg1z 12569	-1 is an integer. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℤ
Theorem	znegclb 12570	A complex number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℤ ↔ -𝐴 ∈ ℤ))
Theorem	nn0negz 12571	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ)
Theorem	nn0negzi 12572	The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ -𝑁 ∈ ℤ
Theorem	zaddcl 12573	Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
Theorem	peano2z 12574	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ)
Theorem	zsubcl 12575	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
Theorem	peano2zm 12576	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
Theorem	zletr 12577	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿))
Theorem	zrevaddcl 12578	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ))
Theorem	znnsub 12579	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 12230.) (Contributed by NM, 11-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
Theorem	znn0sub 12580	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 12492.) (Contributed by NM, 14-Jul-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0))
Theorem	nzadd 12581	The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ))
Theorem	zmulcl 12582	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)
Theorem	zltp1le 12583	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
Theorem	zleltp1 12584	Integer ordering relation. (Contributed by NM, 10-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1)))
Theorem	zlem1lt 12585	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	zltlem1 12586	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	zltlem1d 12587	Integer ordering relation, a deduction version. (Contributed by metakunt, 23-May-2024.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	zgt0ge1 12588	An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
⊢ (𝑍 ∈ ℤ → (0 < 𝑍 ↔ 1 ≤ 𝑍))
Theorem	nnleltp1 12589	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ≤ 𝐵 ↔ 𝐴 < (𝐵 + 1)))
Theorem	nnltp1le 12590	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))
Theorem	nnaddm1cl 12591	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) − 1) ∈ ℕ)
Theorem	nn0ltp1le 12592	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
Theorem	nn0leltp1 12593	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1)))
Theorem	nn0ltlem1 12594	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	nn0sub2 12595	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	nn0lt10b 12596	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
Theorem	nn0lt2 12597	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1))
Theorem	nn0le2is012 12598	A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))
Theorem	nn0lem1lt 12599	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnlem1lt 12600	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))