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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0mulcld 8701 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴 · 𝐵) ∈ ℕ0)
 
Theoremnn0readdcl 8702 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ)
 
Theoremnn0ge2m1nn 8703 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)
 
Theoremnn0ge2m1nn0 8704 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)
 
Theoremnn0nndivcl 8705 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℕ0𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ)
 
3.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. 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 *, see df-xr 7505.

 
Syntaxcxnn0 8706 The set of extended nonnegative integers.
class 0*
 
Definitiondf-xnn0 8707 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers *, see df-xr 7505. If we assumed excluded middle, this would be essentially the same as as defined at df-nninf 6770 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
0* = (ℕ0 ∪ {+∞})
 
Theoremelxnn0 8708 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
 
Theoremnn0ssxnn0 8709 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
0 ⊆ ℕ0*
 
Theoremnn0xnn0 8710 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ∈ ℕ0*)
 
Theoremxnn0xr 8711 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ ℝ*)
 
Theorem0xnn0 8712 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
0 ∈ ℕ0*
 
Theorempnf0xnn0 8713 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
+∞ ∈ ℕ0*
 
Theoremnn0nepnf 8714 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ≠ +∞)
 
Theoremnn0xnn0d 8715 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℕ0*)
 
Theoremnn0nepnfd 8716 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ≠ +∞)
 
Theoremxnn0nemnf 8717 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ≠ -∞)
 
Theoremxnn0xrnemnf 8718 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
 
Theoremxnn0nnn0pnf 8719 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 = +∞)
 
3.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 8720 Extend class notation to include the class of integers.
class
 
Definitiondf-z 8721 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)}
 
Theoremelz 8722 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 
Theoremnnnegz 8723 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
 
Theoremzre 8724 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
 
Theoremzcn 8725 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
 
Theoremzrei 8726 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
𝐴 ∈ ℤ       𝐴 ∈ ℝ
 
Theoremzssre 8727 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 8728 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 8729 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℤ ∈ V
 
Theoremelnnz 8730 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
 
Theorem0z 8731 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0 ∈ ℤ
 
Theorem0zd 8732 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℤ)
 
Theoremelnn0z 8733 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
 
Theoremelznn0nn 8734 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
 
Theoremelznn0 8735 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
 
Theoremelznn 8736 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)))
 
Theoremnnssz 8737 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
ℕ ⊆ ℤ
 
Theoremnn0ssz 8738 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
0 ⊆ ℤ
 
Theoremnnz 8739 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
 
Theoremnn0z 8740 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0𝑁 ∈ ℤ)
 
Theoremnnzi 8741 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ       𝑁 ∈ ℤ
 
Theoremnn0zi 8742 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       𝑁 ∈ ℤ
 
Theoremelnnz1 8743 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))
 
Theoremnnzrab 8744 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥}
 
Theoremnn0zrab 8745 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥}
 
Theorem1z 8746 One is an integer. (Contributed by NM, 10-May-2004.)
1 ∈ ℤ
 
Theorem1zzd 8747 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℤ)
 
Theorem2z 8748 Two is an integer. (Contributed by NM, 10-May-2004.)
2 ∈ ℤ
 
Theorem3z 8749 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ ℤ
 
Theorem4z 8750 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
4 ∈ ℤ
 
Theoremznegcl 8751 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
 
Theoremneg1z 8752 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℤ
 
Theoremznegclb 8753 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℤ ↔ -𝐴 ∈ ℤ))
 
Theoremnn0negz 8754 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ)
 
Theoremnn0negzi 8755 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       -𝑁 ∈ ℤ
 
Theorempeano2z 8756 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
(𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ)
 
Theoremzaddcllempos 8757 Lemma for zaddcl 8760. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
Theorempeano2zm 8758 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
(𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
 
Theoremzaddcllemneg 8759 Lemma for zaddcl 8760. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
Theoremzaddcl 8760 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)
 
Theoremzsubcl 8761 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁) ∈ ℤ)
 
Theoremztri3or0 8762 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
(𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
 
Theoremztri3or 8763 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 = 𝑁𝑁 < 𝑀))
 
Theoremzletric 8764 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵𝐴))
 
Theoremzlelttric 8765 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵 < 𝐴))
 
Theoremzltnle 8766 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremzleloe 8767 Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐴 = 𝐵)))
 
Theoremznnnlt1 8768 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))
 
Theoremzletr 8769 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽𝐾𝐾𝐿) → 𝐽𝐿))
 
Theoremzrevaddcl 8770 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
(𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ))
 
Theoremznnsub 8771 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8432.) (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁𝑀) ∈ ℕ))
 
Theoremnzadd 8772 The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ))
 
Theoremzmulcl 8773 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)
 
Theoremzltp1le 8774 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))
 
Theoremzleltp1 8775 Integer ordering relation. (Contributed by NM, 10-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremzlem1lt 8776 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremzltlem1 8777 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremzgt0ge1 8778 An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
(𝑍 ∈ ℤ → (0 < 𝑍 ↔ 1 ≤ 𝑍))
 
Theoremnnleltp1 8779 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴𝐵𝐴 < (𝐵 + 1)))
 
Theoremnnltp1le 8780 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))
 
Theoremnnaddm1cl 8781 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)
 
Theoremnn0ltp1le 8782 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
 
Theoremnn0leltp1 8783 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremnn0ltlem1 8784 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremznn0sub 8785 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8786.) (Contributed by NM, 14-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0sub 8786 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0n0n1ge2 8787 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 ∈ ℕ0𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)
 
Theoremelz2 8788* 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 = (𝑥𝑦))
 
Theoremdfz2 8789 Alternate definition of the integers, based on elz2 8788. (Contributed by Mario Carneiro, 16-May-2014.)
ℤ = ( − “ (ℕ × ℕ))
 
Theoremnn0sub2 8790 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁𝑀) ∈ ℕ0)
 
Theoremzapne 8791 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁𝑀𝑁))
 
Theoremzdceq 8792 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
 
Theoremzdcle 8793 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴𝐵)
 
Theoremzdclt 8794 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
 
Theoremzltlen 8795 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8083 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremnn0n0n1ge2b 8796 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))
 
Theoremnn0lt10b 8797 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
 
Theoremnn0lt2 8798 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
((𝑁 ∈ ℕ0𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1))
 
Theoremnn0lem1lt 8799 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnlem1lt 8800 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
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