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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempnf0xnn0 9001 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
+∞ ∈ ℕ0*
 
Theoremnn0nepnf 9002 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0𝐴 ≠ +∞)
 
Theoremnn0xnn0d 9003 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ∈ ℕ0*)
 
Theoremnn0nepnfd 9004 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
(𝜑𝐴 ∈ ℕ0)       (𝜑𝐴 ≠ +∞)
 
Theoremxnn0nemnf 9005 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ≠ -∞)
 
Theoremxnn0xrnemnf 9006 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
 
Theoremxnn0nnn0pnf 9007 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 = +∞)
 
4.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 9008 Extend class notation to include the class of integers.
class
 
Definitiondf-z 9009 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 9010 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 
Theoremnnnegz 9011 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
 
Theoremzre 9012 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
 
Theoremzcn 9013 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
 
Theoremzrei 9014 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
𝐴 ∈ ℤ       𝐴 ∈ ℝ
 
Theoremzssre 9015 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 9016 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 9017 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℤ ∈ V
 
Theoremelnnz 9018 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
 
Theorem0z 9019 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0 ∈ ℤ
 
Theorem0zd 9020 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℤ)
 
Theoremelnn0z 9021 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
 
Theoremelznn0nn 9022 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 9023 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
 
Theoremelznn 9024 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 9025 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
ℕ ⊆ ℤ
 
Theoremnn0ssz 9026 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
0 ⊆ ℤ
 
Theoremnnz 9027 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
 
Theoremnn0z 9028 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0𝑁 ∈ ℤ)
 
Theoremnnzi 9029 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ       𝑁 ∈ ℤ
 
Theoremnn0zi 9030 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       𝑁 ∈ ℤ
 
Theoremelnnz1 9031 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 9032 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥}
 
Theoremnn0zrab 9033 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥}
 
Theorem1z 9034 One is an integer. (Contributed by NM, 10-May-2004.)
1 ∈ ℤ
 
Theorem1zzd 9035 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℤ)
 
Theorem2z 9036 Two is an integer. (Contributed by NM, 10-May-2004.)
2 ∈ ℤ
 
Theorem3z 9037 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ ℤ
 
Theorem4z 9038 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
4 ∈ ℤ
 
Theoremznegcl 9039 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
 
Theoremneg1z 9040 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℤ
 
Theoremznegclb 9041 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℤ ↔ -𝐴 ∈ ℤ))
 
Theoremnn0negz 9042 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ)
 
Theoremnn0negzi 9043 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       -𝑁 ∈ ℤ
 
Theorempeano2z 9044 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
(𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ)
 
Theoremzaddcllempos 9045 Lemma for zaddcl 9048. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
Theorempeano2zm 9046 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
(𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
 
Theoremzaddcllemneg 9047 Lemma for zaddcl 9048. 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 9048 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 9049 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁) ∈ ℤ)
 
Theoremztri3or0 9050 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
(𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
 
Theoremztri3or 9051 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 = 𝑁𝑁 < 𝑀))
 
Theoremzletric 9052 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵𝐴))
 
Theoremzlelttric 9053 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵 < 𝐴))
 
Theoremzltnle 9054 '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 9055 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 9056 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 9057 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽𝐾𝐾𝐿) → 𝐽𝐿))
 
Theoremzrevaddcl 9058 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
(𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ))
 
Theoremznnsub 9059 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8719.) (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁𝑀) ∈ ℕ))
 
Theoremnzadd 9060 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 9061 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)
 
Theoremzltp1le 9062 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 9063 Integer ordering relation. (Contributed by NM, 10-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremzlem1lt 9064 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremzltlem1 9065 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremzgt0ge1 9066 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 9067 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 9068 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))
 
Theoremnnaddm1cl 9069 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 9070 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 9071 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremnn0ltlem1 9072 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 9073 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9074.) (Contributed by NM, 14-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0sub 9074 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0n0n1ge2 9075 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 9076* 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 9077 Alternate definition of the integers, based on elz2 9076. (Contributed by Mario Carneiro, 16-May-2014.)
ℤ = ( − “ (ℕ × ℕ))
 
Theoremnn0sub2 9078 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁𝑀) ∈ ℕ0)
 
Theoremzapne 9079 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁𝑀𝑁))
 
Theoremzdceq 9080 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
 
Theoremzdcle 9081 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴𝐵)
 
Theoremzdclt 9082 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
 
Theoremzltlen 9083 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8357 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremnn0n0n1ge2b 9084 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 9085 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
 
Theoremnn0lt2 9086 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))
 
Theoremnn0le2is012 9087 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 ∈ ℕ0𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))
 
Theoremnn0lem1lt 9088 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnlem1lt 9089 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnltlem1 9090 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremnnm1ge0 9091 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
(𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1))
 
Theoremnn0ge0div 9092 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℕ0𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿))
 
Theoremzdiv 9093* Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
 
Theoremzdivadd 9094 Property of divisibility: if 𝐷 divides 𝐴 and 𝐵 then it divides 𝐴 + 𝐵. (Contributed by NM, 3-Oct-2008.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 / 𝐷) ∈ ℤ ∧ (𝐵 / 𝐷) ∈ ℤ)) → ((𝐴 + 𝐵) / 𝐷) ∈ ℤ)
 
Theoremzdivmul 9095 Property of divisibility: if 𝐷 divides 𝐴 then it divides 𝐵 · 𝐴. (Contributed by NM, 3-Oct-2008.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 / 𝐷) ∈ ℤ) → ((𝐵 · 𝐴) / 𝐷) ∈ ℤ)
 
Theoremzextle 9096* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘𝑀𝑘𝑁)) → 𝑀 = 𝑁)
 
Theoremzextlt 9097* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 < 𝑀𝑘 < 𝑁)) → 𝑀 = 𝑁)
 
Theoremrecnz 9098 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → ¬ (1 / 𝐴) ∈ ℤ)
 
Theorembtwnnz 9099 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵𝐵 < (𝐴 + 1)) → ¬ 𝐵 ∈ ℤ)
 
Theoremgtndiv 9100 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐵 < 𝐴) → ¬ (𝐵 / 𝐴) ∈ ℤ)
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