P. 106 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fz01or 10501	An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1))
4.5.5  Finite intervals of nonnegative integers Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: (0...𝑁), usually abbreviated by "fz0".
Theorem	elfz2nn0 10502	Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁))
Theorem	fznn0 10503	Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfznn0 10504	A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
Theorem	elfz3nn0 10505	The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
Theorem	fz0ssnn0 10506	Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
⊢ (0...𝑁) ⊆ ℕ0
Theorem	fz1ssfz0 10507	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (1...𝑁) ⊆ (0...𝑁)
Theorem	0elfz 10508	0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
Theorem	nn0fz0 10509	A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
Theorem	elfz0add 10510	An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))
Theorem	fz0sn 10511	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
⊢ (0...0) = {0}
Theorem	fz0tp 10512	An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
⊢ (0...2) = {0, 1, 2}
Theorem	fz0to3un2pr 10513	An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.)
⊢ (0...3) = ({0, 1} ∪ {2, 3})
Theorem	fz0to4untppr 10514	An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})
Theorem	elfz0ubfz0 10515	An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → 𝐾 ∈ (0...𝐿))
Theorem	elfz0fzfz0 10516	A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → 𝑀 ∈ (0...𝑁))
Theorem	fz0fzelfz0 10517	If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))
Theorem	fznn0sub2 10518	Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
Theorem	uzsubfz0 10519	Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − 𝐿) ∈ (0...𝑁))
Theorem	fz0fzdiffz0 10520	The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 − 𝑀) ∈ (0...𝑁))
Theorem	elfzmlbm 10521	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))
Theorem	elfzmlbp 10522	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑀 + 𝑁))) → (𝐾 − 𝑀) ∈ (0...𝑁))
Theorem	fzctr 10523	Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...(2 · 𝑁)))
Theorem	difelfzle 10524	The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑀) → (𝑀 − 𝐾) ∈ (0...𝑁))
Theorem	difelfznle 10525	The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑀) → ((𝑀 + 𝑁) − 𝐾) ∈ (0...𝑁))
Theorem	nn0split 10526	Express the set of nonnegative integers as the disjoint (see nn0disj 10528) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
Theorem	nnsplit 10527	Express the set of positive integers as the disjoint union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
Theorem	nn0disj 10528	The first 𝑁 + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
⊢ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅
Theorem	1fv 10529	A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))
Theorem	4fvwrd4 10530*	The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
⊢ ((𝐿 ∈ (ℤ≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
Theorem	2ffzeq 10531*	Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
4.5.6  Half-open integer ranges
Syntax	cfzo 10532	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10533*	Define a function generating sets of integers using a half-open range. Read (𝑀..^𝑁) as the integers from 𝑀 up to, but not including, 𝑁; contrast with (𝑀...𝑁) df-fz 10395, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10569 with fzsplit 10439, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
Theorem	fzof 10534	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ
Theorem	elfzoel1 10535	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
Theorem	elfzoel2 10536	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
Theorem	elfzoelz 10537	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
Theorem	fzoval 10538	Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
Theorem	elfzo 10539	Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)))
Theorem	elfzo2 10540	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
Theorem	elfzouz 10541	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
Theorem	nelfzo 10542	An integer not being a member of a half-open finite set of integers. (Contributed by AV, 29-Apr-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∉ (𝑀..^𝑁) ↔ (𝐾 < 𝑀 ∨ 𝑁 ≤ 𝐾)))
Theorem	fzodcel 10543	Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzolb 10544	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
Theorem	fzolb2 10545	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
Theorem	elfzole1 10546	A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝐾)
Theorem	elfzolt2 10547	A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁)
Theorem	elfzolt3 10548	Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 < 𝑁)
Theorem	elfzolt2b 10549	A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝐾..^𝑁))
Theorem	elfzolt3b 10550	Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ (𝑀..^𝑁))
Theorem	fzonel 10551	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ ¬ 𝐵 ∈ (𝐴..^𝐵)
Theorem	elfzouz2 10552	The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
Theorem	elfzofz 10553	A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfzo3 10554	Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. 𝐾 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝐾, 𝐾 ∈ (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ (𝐾..^𝑁)))
Theorem	fzom 10555*	A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁))
Theorem	fzossfz 10556	A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵)
Theorem	fzon 10557	A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
Theorem	fzo0n 10558	A half-open range of nonnegative integers is empty iff the upper bound is not positive. (Contributed by AV, 2-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 𝑀 ↔ (0..^(𝑁 − 𝑀)) = ∅))
Theorem	fzonlt0 10559	A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 < 𝑁 ↔ (𝑀..^𝑁) = ∅))
Theorem	fzo0 10560	Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴..^𝐴) = ∅
Theorem	fzonnsub 10561	If 𝐾 < 𝑁 then 𝑁 − 𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝐾) ∈ ℕ)
Theorem	fzonnsub2 10562	If 𝑀 < 𝑁 then 𝑁 − 𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝑀) ∈ ℕ)
Theorem	fzoss1 10563	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁))
Theorem	fzoss2 10564	Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁))
Theorem	fzossrbm1 10565	Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
Theorem	fzo0ss1 10566	Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
⊢ (1..^𝑁) ⊆ (0..^𝑁)
Theorem	fzossnn0 10567	A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
⊢ (𝑀 ∈ ℕ0 → (𝑀..^𝑁) ⊆ ℕ0)
Theorem	fzospliti 10568	One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 ∈ (𝐵..^𝐷) ∨ 𝐴 ∈ (𝐷..^𝐶)))
Theorem	fzosplit 10569	Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐷 ∈ (𝐵...𝐶) → (𝐵..^𝐶) = ((𝐵..^𝐷) ∪ (𝐷..^𝐶)))
Theorem	fzodisj 10570	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
Theorem	fzouzsplit 10571	Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (ℤ≥‘𝐴) = ((𝐴..^𝐵) ∪ (ℤ≥‘𝐵)))
Theorem	fzouzdisj 10572	A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
⊢ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) = ∅
Theorem	fzoun 10573	A half-open integer range as union of two half-open integer ranges. (Contributed by AV, 23-Apr-2022.)
⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℕ0) → (𝐴..^(𝐵 + 𝐶)) = ((𝐴..^𝐵) ∪ (𝐵..^(𝐵 + 𝐶))))
Theorem	fzodisjsn 10574	A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
⊢ ((𝐴..^𝐵) ∩ {𝐵}) = ∅
Theorem	lbfzo0 10575	An integer is strictly greater than zero iff it is a member of ℕ. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
Theorem	elfzo0 10576	Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
Theorem	nn0p1elfzo 10577	A nonnegative integer increased by 1 which is less than or equal to another integer is an element of a half-open range of integers. (Contributed by AV, 27-Feb-2021.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ (𝐾 + 1) ≤ 𝑁) → 𝐾 ∈ (0..^𝑁))
Theorem	fzo1fzo0n0 10578	An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
⊢ (𝐾 ∈ (1..^𝑁) ↔ (𝐾 ∈ (0..^𝑁) ∧ 𝐾 ≠ 0))
Theorem	elfzo0z 10579	Membership in a half-open range of nonnegative integers, generalization of elfzo0 10576 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
Theorem	elfzo0le 10580	A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ (𝐴 ∈ (0..^𝐵) → 𝐴 ≤ 𝐵)
Theorem	elfzonn0 10581	A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ (𝐾 ∈ (0..^𝑁) → 𝐾 ∈ ℕ0)
Theorem	fzonmapblen 10582	The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
⊢ ((𝐴 ∈ (0..^𝑁) ∧ 𝐵 ∈ (0..^𝑁) ∧ 𝐵 < 𝐴) → (𝐵 + (𝑁 − 𝐴)) < 𝑁)
Theorem	fzofzim 10583	If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
⊢ ((𝐾 ≠ 𝑀 ∧ 𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
Theorem	fz1fzo0m1 10584	Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.)
⊢ (𝑀 ∈ (1...𝑁) → (𝑀 − 1) ∈ (0..^𝑁))
Theorem	fzossnn 10585	Half-open integer ranges starting with 1 are subsets of ℕ. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (1..^𝑁) ⊆ ℕ
Theorem	elfzo1 10586	Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀))
Theorem	fzo0m 10587*	A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
Theorem	fzoaddel 10588	Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)..^(𝐶 + 𝐷)))
Theorem	fzo0addel 10589	Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020.)
⊢ ((𝐴 ∈ (0..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ (𝐷..^(𝐶 + 𝐷)))
Theorem	fzo0addelr 10590	Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020.)
⊢ ((𝐴 ∈ (0..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ (𝐷..^(𝐷 + 𝐶)))
Theorem	fzoaddel2 10591	Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (0..^(𝐵 − 𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
Theorem	elfzoextl 10592	Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025.) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025.)
⊢ ((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝐼 + 𝑁)))
Theorem	elfzoext 10593	Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020.) (Proof shortened by AV, 23-Sep-2025.)
⊢ ((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → 𝑍 ∈ (𝑀..^(𝑁 + 𝐼)))
Theorem	elincfzoext 10594	Membership of an increased integer in a correspondingly extended half-open range of integers. (Contributed by AV, 30-Apr-2020.)
⊢ ((𝑍 ∈ (𝑀..^𝑁) ∧ 𝐼 ∈ ℕ0) → (𝑍 + 𝐼) ∈ (𝑀..^(𝑁 + 𝐼)))
Theorem	fzosubel 10595	Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 − 𝐷) ∈ ((𝐵 − 𝐷)..^(𝐶 − 𝐷)))
Theorem	fzosubel2 10596	Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ ((𝐵 + 𝐶)..^(𝐵 + 𝐷)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴 − 𝐵) ∈ (𝐶..^𝐷))
Theorem	fzosubel3 10597	Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (𝐵..^(𝐵 + 𝐷)) ∧ 𝐷 ∈ ℤ) → (𝐴 − 𝐵) ∈ (0..^𝐷))
Theorem	eluzgtdifelfzo 10598	Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ∈ (ℤ≥‘𝐴) ∧ 𝐵 < 𝐴) → (𝑁 − 𝐴) ∈ (0..^(𝑁 − 𝐵))))
Theorem	ige2m2fzo 10599	Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 2) ∈ (0..^(𝑁 − 1)))
Theorem	fzocatel 10600	Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 − 𝐵) ∈ (0..^𝐶))