P. 136 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzn 13501	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
Theorem	fzen 13502	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))
Theorem	fz1n 13503	A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) = ∅ ↔ 𝑁 = 0))
Theorem	0nelfz1 13504	0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.)
⊢ 0 ∉ (1...𝑁)
Theorem	0fz1 13505	Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0))
Theorem	fz10 13506	There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (1...0) = ∅
Theorem	uzsubsubfz 13507	Membership of an integer greater than L decreased by ( L - M ) in an M-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ ((𝐿 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈ (𝑀...𝑁))
Theorem	uzsubsubfz1 13508	Membership of an integer greater than L decreased by ( L - 1 ) in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ ((𝐿 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁))
Theorem	ige3m2fz 13509	Membership of an integer greater than 2 decreased by 2 in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ (1...𝑁))
Theorem	fzsplit2 13510	Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
⊢ (((𝐾 + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
Theorem	fzsplit 13511	Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
Theorem	fzdisj 13512	Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
Theorem	fz01en 13513	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁))
Theorem	elfznn 13514	A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ)
Theorem	elfz1end 13515	A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
Theorem	fz1ssnn 13516	A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (1...𝐴) ⊆ ℕ
Theorem	fznn0sub 13517	Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈ ℕ0)
Theorem	fzmmmeqm 13518	Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀))
Theorem	fzaddel 13519	Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))))
Theorem	fzadd2 13520	Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑂 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → ((𝐽 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑂...𝑃)) → (𝐽 + 𝐾) ∈ ((𝑀 + 𝑂)...(𝑁 + 𝑃))))
Theorem	fzsubel 13521	Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))))
Theorem	fzopth 13522	A finite set of sequential integers has the ordered pair property (compare opth 5436) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
Theorem	fzass4 13523	Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷)))
Theorem	fzss1 13524	Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzss2 13525	Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁))
Theorem	fzssuz 13526	A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
Theorem	fzsn 13527	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
Theorem	fzssp1 13528	Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
Theorem	fzssnn 13529	Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ)
Theorem	ssfzunsnext 13530	A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 13-Nov-2021.)
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
Theorem	ssfzunsn 13531	A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.) (Proof shortened by AV, 13-Nov-2021.)
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
Theorem	fzsuc 13532	Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
Theorem	fzpred 13533	Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
Theorem	fzpreddisj 13534	A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
Theorem	elfzp1 13535	Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
Theorem	fzp1ss 13536	Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzelp1 13537	Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (𝑀...(𝑁 + 1)))
Theorem	fzp1elp1 13538	Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 + 1) ∈ (𝑀...(𝑁 + 1)))
Theorem	fznatpl1 13539	Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁))
Theorem	fzpr 13540	A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
Theorem	fztp 13541	A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)})
Theorem	fz12pr 13542	An integer range between 1 and 2 is a pair. (Contributed by AV, 11-Jan-2023.)
⊢ (1...2) = {1, 2}
Theorem	fzsuc2 13543	Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
Theorem	fzp1disj 13544	(𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.)
⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅
Theorem	fzdifsuc 13545	Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))
Theorem	fzprval 13546*	Two ways of defining the first two values of a sequence on ℕ. (Contributed by NM, 5-Sep-2011.)
⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵))
Theorem	fztpval 13547*	Two ways of defining the first three values of a sequence on ℕ. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥 ∈ (1...3)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵 ∧ (𝐹‘3) = 𝐶))
Theorem	fzrev 13548	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)) ↔ (𝐽 − 𝐾) ∈ (𝑀...𝑁)))
Theorem	fzrev2 13549	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀))))
Theorem	fzrev2i 13550	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)))
Theorem	fzrev3 13551	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ (𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)))
Theorem	fzrev3i 13552	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))
Theorem	fznn 13553	Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁)))
Theorem	elfz1b 13554	Membership in a 1-based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) (Proof shortened by AV, 23-Jan-2022.)
⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))
Theorem	elfz1uz 13555	Membership in a 1-based finite set of sequential integers with an upper integer. (Contributed by AV, 23-Jan-2022.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (1...𝑀))
Theorem	elfzm11 13556	Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)))
Theorem	uzsplit 13557	Express an upper integer set as the disjoint (see uzdisj 13558) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ≥‘𝑁)))
Theorem	uzdisj 13558	The first 𝑁 elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) = ∅
Theorem	fseq1p1m1 13559	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}    ⇒   ⊢ (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))))
Theorem	fseq1m1p1 13560	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ 𝐻 = {⟨𝑁, 𝐵⟩}    ⇒   ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1))))))
Theorem	fz1sbc 13561*	Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ [𝑁 / 𝑘]𝜑))
Theorem	elfzp1b 13562	An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...(𝑁 − 1)) ↔ (𝐾 + 1) ∈ (1...𝑁)))
Theorem	elfzm1b 13563	An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1...𝑁) ↔ (𝐾 − 1) ∈ (0...(𝑁 − 1))))
Theorem	elfzp12 13564	Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))))
Theorem	fzne1 13565	Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁))
Theorem	fzdif1 13566	Split the first element of a finite set of sequential integers. More generic than fzpred 13533. Analogous to fzdif2 32713. (Contributed by AV, 12-Sep-2025.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁))
Theorem	fz0dif1 13567	Split the first element of a finite set of sequential nonnegative integers. (Contributed by AV, 12-Sep-2025.)
⊢ (𝑁 ∈ ℕ0 → ((0...𝑁) ∖ {0}) = (1...𝑁))
Theorem	fzm1 13568	Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁)))
Theorem	fzneuz 13569	No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ≥‘𝐾))
Theorem	fznuz 13570	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
⊢ (𝐾 ∈ (𝑀...𝑁) → ¬ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))
Theorem	uznfz 13571	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
⊢ (𝐾 ∈ (ℤ≥‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))
Theorem	fzp1nel 13572	One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)
Theorem	fzrevral 13573*	Reversal of scanning order inside of a universal quantification restricted to a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))[(𝐾 − 𝑘) / 𝑗]𝜑))
Theorem	fzrevral2 13574*	Reversal of scanning order inside of a universal quantification restricted to a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾 − 𝑘) / 𝑗]𝜑))
Theorem	fzrevral3 13575*	Reversal of scanning order inside of a universal quantification restricted to a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[((𝑀 + 𝑁) − 𝑘) / 𝑗]𝜑))
Theorem	fzshftral 13576*	Shift the scanning order inside of a universal quantification restricted to a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	ige2m1fz1 13577	Membership of an integer greater than 1 decreased by 1 in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ (1...𝑁))
Theorem	ige2m1fz 13578	Membership in a 0-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁))
5.5.6  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 13579	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 13580	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 13581	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 13582	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 13583	Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
⊢ (0...𝑁) ⊆ ℕ0
Theorem	fz1ssfz0 13584	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (1...𝑁) ⊆ (0...𝑁)
Theorem	0elfz 13585	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 13586	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 13587	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 13588	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
⊢ (0...0) = {0}
Theorem	fz0tp 13589	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 13590	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 13591	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.) (Proof shortened by AV, 7-Aug-2025.)
⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})
Theorem	fz0to5un2tp 13592	An integer range from 0 to 5 is the union of two triples. (Contributed by AV, 30-Jul-2025.)
⊢ (0...5) = ({0, 1, 2} ∪ {3, 4, 5})
Theorem	elfz0ubfz0 13593	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 13594	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 13595	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 13596	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 13597	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 13598	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 13599	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 13600	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))