P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elfz2 10401	Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show 𝑀 ∈ ℤ and 𝑁 ∈ ℤ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfzd 10402	Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ≤ 𝐾)    &   ⊢ (𝜑 → 𝐾 ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfz5 10403	Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁))
Theorem	elfz4 10404	Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfzuzb 10405	Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)))
Theorem	eluzfz 10406	Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	elfzuz 10407	A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
Theorem	elfzuz3 10408	Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
Theorem	elfzel2 10409	Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
Theorem	elfzel1 10410	Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
Theorem	elfzelz 10411	A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
Theorem	elfzelzd 10412	A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))    ⇒   ⊢ (𝜑 → 𝐾 ∈ ℤ)
Theorem	fzssz 10413	A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝑀...𝑁) ⊆ ℤ
Theorem	elfzle1 10414	A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
Theorem	elfzle2 10415	A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
Theorem	elfzuz2 10416	Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	elfzle3 10417	Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑁)
Theorem	eluzfz1 10418	Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
Theorem	eluzfz2 10419	Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
Theorem	eluzfz2b 10420	Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
Theorem	elfz3 10421	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
Theorem	elfz1eq 10422	Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
Theorem	elfzubelfz 10423	If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁))
Theorem	peano2fzr 10424	A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
Theorem	fzm 10425*	Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	fztri3or 10426	Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))
Theorem	fzdcel 10427	Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁))
Theorem	fznlem 10428	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅))
Theorem	fzn 10429	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 10430	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 10431	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	0fz1 10432	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 10433	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	fz00m1 10434	There are no integers between 0 and minus 1. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (0...(0 − 1)) = ∅
Theorem	uzsubsubfz 10435	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 10436	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 10437	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 10438	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 10439	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 10440	Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
Theorem	fzsplit3 10441	Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁)))
Theorem	fz01en 10442	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 10443	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 10444	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 10445	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 10446	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 10447	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 10448	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	fzsubel 10449	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 10450	A finite set of sequential integers can represent an ordered pair. (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 10451	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 10452	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 10453	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 10454	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 10455	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
Theorem	fzssp1 10456	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 10457	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	fzsuc 10458	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	fzspl 10459	Split the last element of a finite set of sequential integers. More generic than fzsuc 10458. (Contributed by Thierry Arnoux, 7-Nov-2016.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
Theorem	fzpred 10460	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 10461	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 10462	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 10463	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 10464	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 10465	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 10466	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 10467	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 10468	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	fzsuc2 10469	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 10470	(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 10471	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 10472*	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 10473*	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 10474	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 10475	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 10476	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 10477	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 10478	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 10479	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 10480	Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))
Theorem	elfzm11 10481	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 10482	Express an upper integer set as the disjoint (see uzdisj 10483) 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 10483	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 10484	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 10485	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 10486*	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 10487	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 10488	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 10489	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	fzm1 10490	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 10491	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 10492	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 10493	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 10494	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 10495*	Reversal of scanning order inside of a quantification over 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 10496*	Reversal of scanning order inside of a quantification over 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 10497*	Reversal of scanning order inside of a quantification over 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 10498*	Shift the scanning order inside of a quantification over 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 10499	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 10500	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))