Type  Label  Description 
Statement 

Theorem  ioossico 8601 
An open interval is a subset of its closurebelow. (Contributed by
Thierry Arnoux, 3Mar2017.)

⊢ (A(,)B) ⊆
(A[,)B) 

Theorem  iocssioo 8602 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)

⊢ (((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) ∧ (A ≤ 𝐶 ∧ 𝐷 < B)) → (𝐶(,]𝐷) ⊆ (A(,)B)) 

Theorem  icossioo 8603 
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29Mar2017.)

⊢ (((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) ∧ (A < 𝐶 ∧ 𝐷 ≤ B)) → (𝐶[,)𝐷) ⊆ (A(,)B)) 

Theorem  ioossioo 8604 
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26Sep2017.)

⊢ (((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) ∧ (A ≤ 𝐶 ∧ 𝐷 ≤ B)) → (𝐶(,)𝐷) ⊆ (A(,)B)) 

Theorem  iccsupr 8605* 
A nonempty subset of a closed real interval satisfies the conditions for
the existence of its supremum. To be useful without excluded middle,
we'll probably need to change not equal to apart, and perhaps make other
changes, but the theorem does hold as stated here. (Contributed by Paul
Chapman, 21Jan2008.)

⊢ (((A ∈ ℝ ∧
B ∈
ℝ) ∧ 𝑆 ⊆ (A[,]B) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃x ∈ ℝ ∀y ∈ 𝑆 y ≤
x)) 

Theorem  elioopnf 8606 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)

⊢ (A ∈ ℝ^{*} → (B ∈ (A(,)+∞) ↔ (B ∈ ℝ ∧ A <
B))) 

Theorem  elioomnf 8607 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)

⊢ (A ∈ ℝ^{*} → (B ∈
(∞(,)A) ↔ (B ∈ ℝ ∧ B <
A))) 

Theorem  elicopnf 8608 
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16Sep2014.)

⊢ (A ∈ ℝ → (B ∈ (A[,)+∞) ↔ (B ∈ ℝ ∧ A ≤
B))) 

Theorem  repos 8609 
Two ways of saying that a real number is positive. (Contributed by NM,
7May2007.)

⊢ (A ∈ (0(,)+∞) ↔ (A ∈ ℝ ∧ 0 < A)) 

Theorem  ioof 8610 
The set of open intervals of extended reals maps to subsets of reals.
(Contributed by NM, 7Feb2007.) (Revised by Mario Carneiro,
16Nov2013.)

⊢ (,):(ℝ^{*} ×
ℝ^{*})⟶𝒫 ℝ 

Theorem  iccf 8611 
The set of closed intervals of extended reals maps to subsets of
extended reals. (Contributed by FL, 14Jun2007.) (Revised by Mario
Carneiro, 3Nov2013.)

⊢ [,]:(ℝ^{*} ×
ℝ^{*})⟶𝒫 ℝ^{*} 

Theorem  unirnioo 8612 
The union of the range of the open interval function. (Contributed by
NM, 7May2007.) (Revised by Mario Carneiro, 30Jan2014.)

⊢ ℝ = ∪ ran
(,) 

Theorem  dfioo2 8613* 
Alternate definition of the set of open intervals of extended reals.
(Contributed by NM, 1Mar2007.) (Revised by Mario Carneiro,
1Sep2015.)

⊢ (,) = (x
∈ ℝ^{*}, y ∈
ℝ^{*} ↦ {w ∈ ℝ ∣ (x < w ∧ w <
y)}) 

Theorem  ioorebasg 8614 
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4Apr2020.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) → (A(,)B) ∈ ran (,)) 

Theorem  elrege0 8615 
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2Sep2009.) (Proof shortened by Mario Carneiro, 18Jun2014.)

⊢ (A ∈ (0[,)+∞) ↔ (A ∈ ℝ ∧ 0 ≤ A)) 

Theorem  rge0ssre 8616 
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9Sep2018.) (Proof shortened by AV, 8Sep2019.)

⊢ (0[,)+∞) ⊆ ℝ 

Theorem  elxrge0 8617 
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28Jun2014.)

⊢ (A ∈ (0[,]+∞) ↔ (A ∈
ℝ^{*} ∧ 0 ≤ A)) 

Theorem  0e0icopnf 8618 
0 is a member of (0[,)+∞) (common case).
(Contributed by David
A. Wheeler, 8Dec2018.)

⊢ 0 ∈
(0[,)+∞) 

Theorem  0e0iccpnf 8619 
0 is a member of (0[,]+∞) (common case).
(Contributed by David
A. Wheeler, 8Dec2018.)

⊢ 0 ∈
(0[,]+∞) 

Theorem  ge0addcl 8620 
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19Jun2014.)

⊢ ((A ∈ (0[,)+∞) ∧ B ∈ (0[,)+∞)) → (A + B) ∈ (0[,)+∞)) 

Theorem  ge0mulcl 8621 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)

⊢ ((A ∈ (0[,)+∞) ∧ B ∈ (0[,)+∞)) → (A · B)
∈ (0[,)+∞)) 

Theorem  lbicc2 8622 
The lower bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.) (Revised by
Mario Carneiro, 9Sep2015.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ A ≤
B) → A ∈ (A[,]B)) 

Theorem  ubicc2 8623 
The upper bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ A ≤
B) → B ∈ (A[,]B)) 

Theorem  0elunit 8624 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 0 ∈
(0[,]1) 

Theorem  1elunit 8625 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 1 ∈
(0[,]1) 

Theorem  iooneg 8626 
Membership in a negated open real interval. (Contributed by Paul Chapman,
26Nov2007.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝐶 ∈ (A(,)B) ↔
𝐶 ∈ (B(,)A))) 

Theorem  iccneg 8627 
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26Nov2007.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝐶 ∈ (A[,]B) ↔
𝐶 ∈ (B[,]A))) 

Theorem  icoshft 8628 
A shifted real is a member of a shifted, closedbelow, openabove real
interval. (Contributed by Paul Chapman, 25Mar2008.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝑋 ∈ (A[,)B) →
(𝑋 + 𝐶) ∈
((A + 𝐶)[,)(B
+ 𝐶)))) 

Theorem  icoshftf1o 8629* 
Shifting a closedbelow, openabove interval is onetoone onto.
(Contributed by Paul Chapman, 25Mar2008.) (Proof shortened by Mario
Carneiro, 1Sep2015.)

⊢ 𝐹 = (x
∈ (A[,)B) ↦
(x + 𝐶)) ⇒ ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐹:(A[,)B)–11onto→((A +
𝐶)[,)(B + 𝐶))) 

Theorem  icodisj 8630 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ 𝐶 ∈
ℝ^{*}) → ((A[,)B) ∩ (B[,)𝐶)) = ∅) 

Theorem  ioodisj 8631 
If the upper bound of one open interval is less than or equal to the
lower bound of the other, the intervals are disjoint. (Contributed by
Jeff Hankins, 13Jul2009.)

⊢ ((((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) ∧ (𝐶 ∈
ℝ^{*} ∧ 𝐷 ∈
ℝ^{*})) ∧ B ≤ 𝐶) → ((A(,)B) ∩
(𝐶(,)𝐷)) = ∅) 

Theorem  iccshftr 8632 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A + 𝑅) = 𝐶
& ⊢ (B + 𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈
(A[,]B) ↔ (𝑋 + 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccshftri 8633 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ
& ⊢ (A + 𝑅) = 𝐶
& ⊢ (B + 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 + 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  iccshftl 8634 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A −
𝑅) = 𝐶
& ⊢ (B −
𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈
(A[,]B) ↔ (𝑋 − 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccshftli 8635 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ
& ⊢ (A −
𝑅) = 𝐶
& ⊢ (B −
𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 − 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  iccdil 8636 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A ·
𝑅) = 𝐶
& ⊢ (B ·
𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ^{+})) → (𝑋 ∈
(A[,]B) ↔ (𝑋 · 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccdili 8637 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ^{+}
& ⊢ (A ·
𝑅) = 𝐶
& ⊢ (B ·
𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 · 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  icccntr 8638 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A / 𝑅) = 𝐶
& ⊢ (B / 𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ^{+})) → (𝑋 ∈
(A[,]B) ↔ (𝑋 / 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  icccntri 8639 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ^{+}
& ⊢ (A / 𝑅) = 𝐶
& ⊢ (B / 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 / 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  divelunit 8640 
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11Jun2013.)

⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ 0 < B)) → ((A
/ B) ∈
(0[,]1) ↔ A ≤ B)) 

Theorem  lincmb01cmp 8641 
A linear combination of two reals which lies in the interval between
them. (Contributed by Jeff Madsen, 2Sep2009.) (Proof shortened by
Mario Carneiro, 8Sep2015.)

⊢ (((A ∈ ℝ ∧
B ∈
ℝ ∧ A < B) ∧ 𝑇 ∈
(0[,]1)) → (((1 − 𝑇) · A) + (𝑇 · B)) ∈ (A[,]B)) 

Theorem  iccf1o 8642* 
Describe a bijection from [0, 1] to an arbitrary
nontrivial
closed interval [A,
B]. (Contributed by Mario Carneiro,
8Sep2015.)

⊢ 𝐹 = (x
∈ (0[,]1) ↦ ((x · B) +
((1 − x) · A))) ⇒ ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧
A < B) → (𝐹:(0[,]1)–11onto→(A[,]B) ∧ ^{◡}𝐹 = (y ∈ (A[,]B) ↦
((y − A) / (B −
A))))) 

Theorem  unitssre 8643 
(0[,]1) is a subset of the reals. (Contributed by
David Moews,
28Feb2017.)

⊢ (0[,]1) ⊆ ℝ 

3.5.4 Finite intervals of integers


Syntax  cfz 8644 
Extend class notation to include the notation for a contiguous finite set
of integers. Read "𝑀...𝑁 " as "the set of integers
from 𝑀 to
𝑁 inclusive."

class ... 

Definition  dffz 8645* 
Define an operation that produces a finite set of sequential integers.
Read "𝑀...𝑁 " as "the set of integers
from 𝑀 to 𝑁
inclusive." See fzval 8646 for its value and additional comments.
(Contributed by NM, 6Sep2005.)

⊢ ... = (𝑚 ∈
ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ
∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) 

Theorem  fzval 8646* 
The value of a finite set of sequential integers. E.g., 2...5
means the set {2, 3, 4, 5}. A special case of
this definition
(starting at 1) appears as Definition 112.1 of [Gleason] p. 141, where
ℕ_k means our 1...𝑘; he calls these sets
segments of the
integers. (Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
3Nov2013.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ
∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) 

Theorem  fzval2 8647 
An alternative way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3Nov2013.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ)) 

Theorem  fzf 8648 
Establish the domain and codomain of the finite integer sequence
function. (Contributed by Scott Fenton, 8Aug2013.) (Revised by Mario
Carneiro, 16Nov2013.)

⊢ ...:(ℤ × ℤ)⟶𝒫
ℤ 

Theorem  elfz1 8649 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈
ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz 8650 
Membership in a finite set of sequential integers. (Contributed by NM,
29Sep2005.)

⊢ ((𝐾 ∈
ℤ ∧ 𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz2 8651 
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, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ 𝐾 ∈
ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz5 8652 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)

⊢ ((𝐾 ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) 

Theorem  elfz4 8653 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ (((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ 𝐾 ∈
ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  elfzuzb 8654 
Membership in a finite set of sequential integers in terms of sets of
upper integers. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾))) 

Theorem  eluzfz 8655 
Membership in a finite set of sequential integers. (Contributed by NM,
4Oct2005.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ ((𝐾 ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  elfzuz 8656 
A member of a finite set of sequential integers belongs to an upper set of
integers. (Contributed by NM, 17Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈
(ℤ_{≥}‘𝑀)) 

Theorem  elfzuz3 8657 
Membership in a finite set of sequential integers implies membership in an
upper set of integers. (Contributed by NM, 28Sep2005.) (Revised by
Mario Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈
(ℤ_{≥}‘𝐾)) 

Theorem  elfzel2 8658 
Membership in a finite set of sequential integer implies the upper bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈
ℤ) 

Theorem  elfzel1 8659 
Membership in a finite set of sequential integer implies the lower bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈
ℤ) 

Theorem  elfzelz 8660 
A member of a finite set of sequential integer is an integer.
(Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈
ℤ) 

Theorem  elfzle1 8661 
A member of a finite set of sequential integer is greater than or equal to
the lower bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾) 

Theorem  elfzle2 8662 
A member of a finite set of sequential integer is less than or equal to
the upper bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁) 

Theorem  elfzuz2 8663 
Implication of membership in a finite set of sequential integers.
(Contributed by NM, 20Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈
(ℤ_{≥}‘𝑀)) 

Theorem  elfzle3 8664 
Membership in a finite set of sequential integer implies the bounds are
comparable. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑁) 

Theorem  eluzfz1 8665 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) 

Theorem  eluzfz2 8666 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 13Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) 

Theorem  eluzfz2b 8667 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) 

Theorem  elfz3 8668 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

⊢ (𝑁 ∈
ℤ → 𝑁 ∈ (𝑁...𝑁)) 

Theorem  elfz1eq 8669 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)

⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) 

Theorem  elfzubelfz 8670 
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, 31May2018.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁)) 

Theorem  peano2fzr 8671 
A Peanopostulatelike theorem for downward closure of a finite set of
sequential integers. (Contributed by Mario Carneiro, 27May2014.)

⊢ ((𝐾 ∈
(ℤ_{≥}‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  fzm 8672* 
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15Apr2020.)

⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈
(ℤ_{≥}‘𝑀)) 

Theorem  fztri3or 8673 
Trichotomy in terms of a finite interval of integers. (Contributed by Jim
Kingdon, 1Jun2020.)

⊢ ((𝐾 ∈
ℤ ∧ 𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝐾 <
𝑀
∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾)) 

Theorem  fzdcel 8674 
Decidability of membership in a finite interval of integers. (Contributed
by Jim Kingdon, 1Jun2020.)

⊢ ((𝐾 ∈
ℤ ∧ 𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → DECID 𝐾 ∈ (𝑀...𝑁)) 

Theorem  fznlem 8675 
A finite set of sequential integers is empty if the bounds are
reversed. (Contributed by Jim Kingdon, 16Apr2020.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝑁 <
𝑀 → (𝑀...𝑁) = ∅)) 

Theorem  fzn 8676 
A finite set of sequential integers is empty if the bounds are
reversed. (Contributed by NM, 22Aug2005.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) → (𝑁 <
𝑀 ↔ (𝑀...𝑁) = ∅)) 

Theorem  fzen 8677 
A shifted finite set of sequential integers is equinumerous to the
original set. (Contributed by Paul Chapman, 11Apr2009.)

⊢ ((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ 𝐾 ∈
ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) 

Theorem  fz1n 8678 
A 1based finite set of sequential integers is empty iff it ends at index
0. (Contributed by Paul Chapman, 22Jun2011.)

⊢ (𝑁 ∈
ℕ_{0} → ((1...𝑁) = ∅ ↔ 𝑁 = 0)) 

Theorem  0fz1 8679 
Two ways to say a finite 1based sequence is empty. (Contributed by Paul
Chapman, 26Oct2012.)

⊢ ((𝑁 ∈
ℕ_{0} ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0)) 

Theorem  fz10 8680 
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16Jun2010.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (1...0) = ∅ 

Theorem  uzsubsubfz 8681 
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, 14Sep2018.)

⊢ ((𝐿 ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈
(𝑀...𝑁)) 

Theorem  uzsubsubfz1 8682 
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, 14Sep2018.)

⊢ ((𝐿 ∈
ℕ ∧ 𝑁 ∈
(ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁)) 

Theorem  ige3m2fz 8683 
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,
14Sep2018.)

⊢ (𝑁 ∈
(ℤ_{≥}‘3) → (𝑁 − 2) ∈ (1...𝑁)) 

Theorem  fzsplit2 8684 
Split a finite interval of integers into two parts. (Contributed by
Mario Carneiro, 13Apr2016.)

⊢ (((𝐾 + 1) ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) 

Theorem  fzsplit 8685 
Split a finite interval of integers into two parts. (Contributed by
Jeff Madsen, 17Jun2010.) (Revised by Mario Carneiro, 13Apr2016.)

⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) 

Theorem  fzdisj 8686 
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17Jun2010.)

⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) 

Theorem  fz01en 8687 
0based and 1based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11Apr2009.)

⊢ (𝑁 ∈
ℤ → (0...(𝑁
− 1)) ≈ (1...𝑁)) 

Theorem  elfznn 8688 
A member of a finite set of sequential integers starting at 1 is a
positive integer. (Contributed by NM, 24Aug2005.)

⊢ (𝐾 ∈
(1...𝑁) → 𝐾 ∈ ℕ) 

Theorem  elfz1end 8689 
A nonempty finite range of integers contains its end point. (Contributed
by Stefan O'Rear, 10Oct2014.)

⊢ (A ∈ ℕ ↔ A ∈ (1...A)) 

Theorem  fznn0sub 8690 
Subtraction closure for a member of a finite set of sequential integers.
(Contributed by NM, 16Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈
ℕ_{0}) 

Theorem  fzmmmeqm 8691 
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,
27May2018.)

⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀)) 

Theorem  fzaddel 8692 
Membership of a sum in a finite set of sequential integers. (Contributed
by NM, 30Jul2005.)

⊢ (((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) ∧ (𝐽 ∈
ℤ ∧ 𝐾 ∈
ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈
((𝑀 + 𝐾)...(𝑁 + 𝐾)))) 

Theorem  fzsubel 8693 
Membership of a difference in a finite set of sequential integers.
(Contributed by NM, 30Jul2005.)

⊢ (((𝑀 ∈
ℤ ∧ 𝑁 ∈
ℤ) ∧ (𝐽 ∈
ℤ ∧ 𝐾 ∈
ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈
((𝑀 − 𝐾)...(𝑁 − 𝐾)))) 

Theorem  fzopth 8694 
A finite set of sequential integers can represent an ordered pair.
(Contributed by NM, 31Oct2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) 

Theorem  fzass4 8695 
Two ways to express a nondecreasing sequence of four integers.
(Contributed by Stefan O'Rear, 15Aug2015.)

⊢ ((B ∈ (A...𝐷) ∧ 𝐶 ∈
(B...𝐷)) ↔ (B ∈ (A...𝐶) ∧ 𝐶 ∈ (A...𝐷))) 

Theorem  fzss1 8696 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 28Sep2005.) (Proof shortened by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈
(ℤ_{≥}‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) 

Theorem  fzss2 8697 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 4Oct2005.) (Revised by Mario Carneiro,
30Apr2015.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁)) 

Theorem  fzssuz 8698 
A finite set of sequential integers is a subset of an upper set of
integers. (Contributed by NM, 28Oct2005.)

⊢ (𝑀...𝑁) ⊆
(ℤ_{≥}‘𝑀) 

Theorem  fzsn 8699 
A finite interval of integers with one element. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈
ℤ → (𝑀...𝑀) = {𝑀}) 

Theorem  fzssp1 8700 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) 