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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxnn0nnn0pnf 8501 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 8502 Extend class notation to include the class of integers.
class
 
Definitiondf-z 8503 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 8504 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 
Theoremnnnegz 8505 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
(𝑁 ∈ ℕ → -𝑁 ∈ ℤ)
 
Theoremzre 8506 An integer is a real. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
 
Theoremzcn 8507 An integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
 
Theoremzrei 8508 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
𝐴 ∈ ℤ       𝐴 ∈ ℝ
 
Theoremzssre 8509 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℝ
 
Theoremzsscn 8510 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℤ ⊆ ℂ
 
Theoremzex 8511 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℤ ∈ V
 
Theoremelnnz 8512 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
 
Theorem0z 8513 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
0 ∈ ℤ
 
Theorem0zd 8514 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℤ)
 
Theoremelnn0z 8515 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁))
 
Theoremelznn0nn 8516 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 8517 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)))
 
Theoremelznn 8518 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 8519 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
ℕ ⊆ ℤ
 
Theoremnn0ssz 8520 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
0 ⊆ ℤ
 
Theoremnnz 8521 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
 
Theoremnn0z 8522 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0𝑁 ∈ ℤ)
 
Theoremnnzi 8523 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ       𝑁 ∈ ℤ
 
Theoremnn0zi 8524 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       𝑁 ∈ ℤ
 
Theoremelnnz1 8525 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 8526 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥}
 
Theoremnn0zrab 8527 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥}
 
Theorem1z 8528 One is an integer. (Contributed by NM, 10-May-2004.)
1 ∈ ℤ
 
Theorem1zzd 8529 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℤ)
 
Theorem2z 8530 Two is an integer. (Contributed by NM, 10-May-2004.)
2 ∈ ℤ
 
Theorem3z 8531 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ ℤ
 
Theorem4z 8532 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
4 ∈ ℤ
 
Theoremznegcl 8533 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
 
Theoremneg1z 8534 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℤ
 
Theoremznegclb 8535 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℤ ↔ -𝐴 ∈ ℤ))
 
Theoremnn0negz 8536 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ)
 
Theoremnn0negzi 8537 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ0       -𝑁 ∈ ℤ
 
Theorempeano2z 8538 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
(𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ)
 
Theoremzaddcllempos 8539 Lemma for zaddcl 8542. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
Theorempeano2zm 8540 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
(𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
 
Theoremzaddcllemneg 8541 Lemma for zaddcl 8542. 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 8542 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 8543 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁) ∈ ℤ)
 
Theoremztri3or0 8544 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
(𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
 
Theoremztri3or 8545 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 = 𝑁𝑁 < 𝑀))
 
Theoremzletric 8546 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵𝐴))
 
Theoremzlelttric 8547 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴𝐵𝐵 < 𝐴))
 
Theoremzltnle 8548 '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 8549 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 8550 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 8551 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽𝐾𝐾𝐿) → 𝐽𝐿))
 
Theoremzrevaddcl 8552 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
(𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ))
 
Theoremznnsub 8553 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8214.) (Contributed by NM, 11-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁𝑀) ∈ ℕ))
 
Theoremnzadd 8554 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 8555 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)
 
Theoremzltp1le 8556 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 8557 Integer ordering relation. (Contributed by NM, 10-May-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremzlem1lt 8558 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremzltlem1 8559 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremzgt0ge1 8560 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 8561 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 8562 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵))
 
Theoremnnaddm1cl 8563 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 8564 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 8565 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremnn0ltlem1 8566 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 8567 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8568.) (Contributed by NM, 14-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0sub 8568 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑁𝑀) ∈ ℕ0))
 
Theoremnn0n0n1ge2 8569 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 8570* 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 8571 Alternate definition of the integers, based on elz2 8570. (Contributed by Mario Carneiro, 16-May-2014.)
ℤ = ( − “ (ℕ × ℕ))
 
Theoremnn0sub2 8572 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝑁𝑀) ∈ ℕ0)
 
Theoremzapne 8573 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁𝑀𝑁))
 
Theoremzdceq 8574 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
 
Theoremzdcle 8575 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴𝐵)
 
Theoremzdclt 8576 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
 
Theoremzltlen 8577 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7867 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremnn0n0n1ge2b 8578 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 8579 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
 
Theoremnn0lt2 8580 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 8581 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnlem1lt 8582 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnltlem1 8583 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremnnm1ge0 8584 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 8585 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 8586* 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 8587 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 8588 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 8589* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘𝑀𝑘𝑁)) → 𝑀 = 𝑁)
 
Theoremzextlt 8590* 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 8591 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 8592 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 8593 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)
 
Theoremhalfnz 8594 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
¬ (1 / 2) ∈ ℤ
 
Theorem3halfnz 8595 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
¬ (3 / 2) ∈ ℤ
 
Theoremsuprzclex 8596* The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℤ)       (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
 
Theoremprime 8597* Two ways to express "𝐴 is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
(𝐴 ∈ ℕ → (∀𝑥 ∈ ℕ ((𝐴 / 𝑥) ∈ ℕ → (𝑥 = 1 ∨ 𝑥 = 𝐴)) ↔ ∀𝑥 ∈ ℕ ((1 < 𝑥𝑥𝐴 ∧ (𝐴 / 𝑥) ∈ ℕ) → 𝑥 = 𝐴)))
 
Theoremmsqznn 8598 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → (𝐴 · 𝐴) ∈ ℕ)
 
Theoremzneo 8599 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ≠ ((2 · 𝐵) + 1))
 
Theoremnneoor 8600 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
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