Type  Label  Description 
Statement 

Theorem  ioossioo 8801 
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 8802* 
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 8803 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)

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

Theorem  elioomnf 8804 
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18Jun2014.)

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

Theorem  elicopnf 8805 
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16Sep2014.)

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

Theorem  repos 8806 
Two ways of saying that a real number is positive. (Contributed by NM,
7May2007.)

⊢ (𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) 

Theorem  ioof 8807 
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 8808 
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 8809 
The union of the range of the open interval function. (Contributed by
NM, 7May2007.) (Revised by Mario Carneiro, 30Jan2014.)

⊢ ℝ = ∪ ran
(,) 

Theorem  dfioo2 8810* 
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 8811 
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 8812 
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 8813 
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9Sep2018.) (Proof shortened by AV, 8Sep2019.)

⊢ (0[,)+∞) ⊆
ℝ 

Theorem  elxrge0 8814 
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28Jun2014.)

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

Theorem  0e0icopnf 8815 
0 is a member of (0[,)+∞) (common case).
(Contributed by David
A. Wheeler, 8Dec2018.)

⊢ 0 ∈ (0[,)+∞) 

Theorem  0e0iccpnf 8816 
0 is a member of (0[,]+∞) (common case).
(Contributed by David
A. Wheeler, 8Dec2018.)

⊢ 0 ∈ (0[,]+∞) 

Theorem  ge0addcl 8817 
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19Jun2014.)

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

Theorem  ge0mulcl 8818 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)

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

Theorem  lbicc2 8819 
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 8820 
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 8821 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 0 ∈ (0[,]1) 

Theorem  1elunit 8822 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 1 ∈ (0[,]1) 

Theorem  iooneg 8823 
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 8824 
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 8825 
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 8826* 
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 8827 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)

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

Theorem  ioodisj 8828 
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 8829 
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 8830 
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 8831 
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 8832 
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 8833 
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 8834 
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 8835 
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 8836 
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 8837 
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 8838 
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 8839* 
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 8840 
(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 8841 
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 8842* 
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 8843 for its value and additional comments.
(Contributed by NM, 6Sep2005.)

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

Theorem  fzval 8843* 
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 8844 
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 8845 
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 8846 
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 8847 
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 8848 
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 8849 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)

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

Theorem  elfz4 8850 
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 8851 
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 8852 
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 8853 
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 8854 
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 8855 
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 8856 
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 8857 
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 8858 
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 8859 
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 8860 
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 8861 
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 8862 
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 8863 
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 8864 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) 

Theorem  elfz3 8865 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) 

Theorem  elfz1eq 8866 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)

⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) 

Theorem  elfzubelfz 8867 
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 8868 
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 8869* 
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15Apr2020.)

⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ_{≥}‘𝑀)) 

Theorem  fztri3or 8870 
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 8871 
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 8872 
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 8873 
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 8874 
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 8875 
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 8876 
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 8877 
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16Jun2010.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (1...0) = ∅ 

Theorem  uzsubsubfz 8878 
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 8879 
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 8880 
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 8881 
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 8882 
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 8883 
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17Jun2010.)

⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) 

Theorem  fz01en 8884 
0based and 1based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11Apr2009.)

⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁)) 

Theorem  elfznn 8885 
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 8886 
A nonempty finite range of integers contains its end point. (Contributed
by Stefan O'Rear, 10Oct2014.)

⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) 

Theorem  fznn0sub 8887 
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 8888 
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 8889 
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 8890 
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 8891 
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 8892 
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 8893 
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 8894 
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 8895 
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 8896 
A finite interval of integers with one element. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) 

Theorem  fzssp1 8897 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) 

Theorem  fzsuc 8898 
Join a successor to the end of a finite set of sequential integers.
(Contributed by NM, 19Jul2008.) (Revised by Mario Carneiro,
28Apr2015.)

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

Theorem  fzpred 8899 
Join a predecessor to the beginning of a finite set of sequential
integers. (Contributed by AV, 24Aug2019.)

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

Theorem  fzpreddisj 8900 
A finite set of sequential integers is disjoint with its predecessor.
(Contributed by AV, 24Aug2019.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) 