P. 138 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzolb 13701	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 alternative 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 13702	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 alternative 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 13703	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 13704	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 13705	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 13706	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 13707	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	elfzop1le2 13708	A member in a half-open integer interval plus 1 is less than or equal to the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 + 1) ≤ 𝑁)
Theorem	fzonel 13709	A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ ¬ 𝐵 ∈ (𝐴..^𝐵)
Theorem	elfzouz2 13710	The upper bound of a half-open range is greater than or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
Theorem	elfzofz 13711	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 13712	Express membership in a half-open integer interval in terms of the "less than or equal to" 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	fzon0 13713	A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ ((𝑀..^𝑁) ≠ ∅ ↔ 𝑀 ∈ (𝑀..^𝑁))
Theorem	fzossfz 13714	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	fzossz 13715	A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝑀..^𝑁) ⊆ ℤ
Theorem	fzon 13716	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 13717	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 13718	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 13719	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 13720	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 13721	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 13722	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 13723	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 13724	Subset of a half-open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
Theorem	fzo0ss1 13725	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 13726	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 13727	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 13728	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 13729	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
Theorem	fzouzsplit 13730	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 13731	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 13732	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 13733	A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.)
⊢ ((𝐴..^𝐵) ∩ {𝐵}) = ∅
Theorem	prinfzo0 13734	The intersection of a half-open integer range and the pair of its outer left borders is empty. (Contributed by AV, 9-Jan-2021.)
⊢ (𝑀 ∈ ℤ → ({𝑀, 𝑁} ∩ ((𝑀 + 1)..^𝑁)) = ∅)
Theorem	lbfzo0 13735	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 13736	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	elfzo0z 13737	Membership in a half-open range of nonnegative integers, generalization of elfzo0 13736 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	nn0p1elfzo 13738	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	elfzo0le 13739	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	elfzolem1 13740	A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1))
Theorem	elfzo0subge1 13741	The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is greater than or equal to 1. (Contributed by AV, 1-Sep-2025.) (Proof shortened by SN, 18-Sep-2025.)
⊢ (𝐴 ∈ (0..^𝐵) → 1 ≤ (𝐵 − 𝐴))
Theorem	elfzo0suble 13742	The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is less than or equal to the upper bound. (Contributed by AV, 1-Sep-2025.) (Proof shortened by SN, 18-Sep-2025.)
⊢ (𝐴 ∈ (0..^𝐵) → (𝐵 − 𝐴) ≤ 𝐵)
Theorem	elfzonn0 13743	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 13744	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 than 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 13745	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 13746	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 13747	Half-open integer ranges starting with 1 are subsets of ℕ. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (1..^𝑁) ⊆ ℕ
Theorem	elfzo1 13748	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	fzo1lb 13749	1 is the left endpoint of a half-open integer range based at 1 iff the right endpoint is an integer greater than 1. (Contributed by AV, 4-Sep-2025.)
⊢ (1 ∈ (1..^𝑁) ↔ 𝑁 ∈ (ℤ≥‘2))
Theorem	fzo1fzo0n0 13750	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	fzo0n0 13751	A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ ((0..^𝐴) ≠ ∅ ↔ 𝐴 ∈ ℕ)
Theorem	fzoaddel 13752	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 13753	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 13754	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 13755	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 13756	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 13757	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 13758	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 13759	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 13760	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 13761	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 13762	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 13763	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 13764	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))
Theorem	ubmelfzo 13765	If an integer in a 1-based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.)
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − 𝐾) ∈ (0..^𝑁))
Theorem	elfzodifsumelfzo 13766	If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.)
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑃)) → (𝐼 ∈ (0..^(𝑁 − 𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃)))
Theorem	elfzom1elp1fzo 13767	Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁))
Theorem	elfzom1elfzo 13768	Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → 𝐼 ∈ (0..^𝑁))
Theorem	fzval3 13769	Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝑁 ∈ ℤ → (𝑀...𝑁) = (𝑀..^(𝑁 + 1)))
Theorem	fz0add1fz1 13770	Translate membership in a 0-based half-open integer range into membership in a 1-based finite sequence of integers. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ (0..^𝑁)) → (𝑋 + 1) ∈ (1...𝑁))
Theorem	fzosn 13771	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
Theorem	elfzomin 13772	Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ (𝑍 ∈ ℤ → 𝑍 ∈ (𝑍..^(𝑍 + 1)))
Theorem	zpnn0elfzo 13773	Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^((𝑍 + 𝑁) + 1)))
Theorem	zpnn0elfzo1 13774	Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1))))
Theorem	fzosplitsnm1 13775	Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}))
Theorem	elfzonlteqm1 13776	If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.)
⊢ ((𝐴 ∈ (0..^𝐵) ∧ ¬ 𝐴 < (𝐵 − 1)) → 𝐴 = (𝐵 − 1))
Theorem	fzonn0p1 13777	A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
Theorem	fzossfzop1 13778	A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
Theorem	fzonn0p1p1 13779	If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
⊢ (𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1)))
Theorem	elfzom1p1elfzo 13780	Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
Theorem	fzo0ssnn0 13781	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) (Proof shortened by JJ, 1-Jun-2021.)
⊢ (0..^𝑁) ⊆ ℕ0
Theorem	fzo01 13782	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (0..^1) = {0}
Theorem	fzo12sn 13783	A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
⊢ (1..^2) = {1}
Theorem	fzo13pr 13784	A 1-based half-open integer interval up to, but not including, 3 is a pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)
⊢ (1..^3) = {1, 2}
Theorem	fzo0to2pr 13785	A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
⊢ (0..^2) = {0, 1}
Theorem	fz01pr 13786	An integer range between 0 and 1 is a pair. (Contributed by AV, 11-Sep-2025.)
⊢ (0...1) = {0, 1}
Theorem	fzo0to3tp 13787	A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
⊢ (0..^3) = {0, 1, 2}
Theorem	fzo0to42pr 13788	A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
⊢ (0..^4) = ({0, 1} ∪ {2, 3})
Theorem	fzo1to4tp 13789	A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.)
⊢ (1..^4) = {1, 2, 3}
Theorem	fzo0sn0fzo1 13790	A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.)
⊢ (𝑁 ∈ ℕ → (0..^𝑁) = ({0} ∪ (1..^𝑁)))
Theorem	elfzo0l 13791	A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021.)
⊢ (𝐾 ∈ (0..^𝑁) → (𝐾 = 0 ∨ 𝐾 ∈ (1..^𝑁)))
Theorem	fzoend 13792	The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
Theorem	fzo0end 13793	The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ (0..^𝐵))
Theorem	ssfzo12 13794	Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	ssfzoulel 13795	If a half-open integer range is a subset of a half-open range of nonnegative integers, but its lower bound is greater than or equal to the upper bound of the containing range, or its upper bound is less than or equal to 0, then its upper bound is less than or equal to its lower bound (and therefore it is actually empty). (Contributed by Alexander van der Vekens, 24-May-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))
Theorem	ssfzo12bi 13796	Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
⊢ (((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	fzoopth 13797	A half-open integer range can represent an ordered pair, analogous to fzopth 13597. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → ((𝑀..^𝑁) = (𝐽..^𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
Theorem	ubmelm1fzo 13798	The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
⊢ (𝐾 ∈ (0..^𝑁) → ((𝑁 − 𝐾) − 1) ∈ (0..^𝑁))
Theorem	fzofzp1 13799	If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐶 ∈ (𝐴..^𝐵) → (𝐶 + 1) ∈ (𝐴...𝐵))
Theorem	fzofzp1b 13800	If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))