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Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremioossicc 9301 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
(𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
 
Theoremicossicc 9302 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
(𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)
 
Theoremiocssicc 9303 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
(𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
 
Theoremioossico 9304 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
(𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
 
Theoremiocssioo 9305 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷 < 𝐵)) → (𝐶(,]𝐷) ⊆ (𝐴(,)𝐵))
 
Theoremicossioo 9306 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐷𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴(,)𝐵))
 
Theoremioossioo 9307 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
 
Theoremiccsupr 9308* 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, 21-Jan-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥))
 
Theoremelioopnf 9309 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
 
Theoremelioomnf 9310 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))
 
Theoremelicopnf 9311 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴𝐵)))
 
Theoremrepos 9312 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
(𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremioof 9313 The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
(,):(ℝ* × ℝ*)⟶𝒫 ℝ
 
Theoremiccf 9314 The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
[,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremunirnioo 9315 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
ℝ = ran (,)
 
Theoremdfioo2 9316* Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)
(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤𝑤 < 𝑦)})
 
Theoremioorebasg 9317 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,))
 
Theoremelrege0 9318 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrge0ssre 9319 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
(0[,)+∞) ⊆ ℝ
 
Theoremelxrge0 9320 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴))
 
Theorem0e0icopnf 9321 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,)+∞)
 
Theorem0e0iccpnf 9322 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,]+∞)
 
Theoremge0addcl 9323 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 + 𝐵) ∈ (0[,)+∞))
 
Theoremge0mulcl 9324 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 · 𝐵) ∈ (0[,)+∞))
 
Theoremlbicc2 9325 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
 
Theoremubicc2 9326 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
 
Theorem0elunit 9327 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
0 ∈ (0[,]1)
 
Theorem1elunit 9328 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
1 ∈ (0[,]1)
 
Theoremiooneg 9329 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴)))
 
Theoremiccneg 9330 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ -𝐶 ∈ (-𝐵[,]-𝐴)))
 
Theoremicoshft 9331 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑋 ∈ (𝐴[,)𝐵) → (𝑋 + 𝐶) ∈ ((𝐴 + 𝐶)[,)(𝐵 + 𝐶))))
 
Theoremicoshftf1o 9332* Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥 ∈ (𝐴[,)𝐵) ↦ (𝑥 + 𝐶))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐹:(𝐴[,)𝐵)–1-1-onto→((𝐴 + 𝐶)[,)(𝐵 + 𝐶)))
 
Theoremicodisj 9333 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
 
Theoremioodisj 9334 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, 13-Jul-2009.)
((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝐵𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅)
 
Theoremiccshftr 9335 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 + 𝑅) = 𝐶    &   (𝐵 + 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷)))
 
Theoremiccshftri 9336 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ    &   (𝐴 + 𝑅) = 𝐶    &   (𝐵 + 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))
 
Theoremiccshftl 9337 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑅) = 𝐶    &   (𝐵𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋𝑅) ∈ (𝐶[,]𝐷)))
 
Theoremiccshftli 9338 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ    &   (𝐴𝑅) = 𝐶    &   (𝐵𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋𝑅) ∈ (𝐶[,]𝐷))
 
Theoremiccdil 9339 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 · 𝑅) = 𝐶    &   (𝐵 · 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)))
 
Theoremiccdili 9340 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ+    &   (𝐴 · 𝑅) = 𝐶    &   (𝐵 · 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))
 
Theoremicccntr 9341 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 / 𝑅) = 𝐶    &   (𝐵 / 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)))
 
Theoremicccntri 9342 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ+    &   (𝐴 / 𝑅) = 𝐶    &   (𝐵 / 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))
 
Theoremdivelunit 9343 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴𝐵))
 
Theoremlincmb01cmp 9344 A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵))
 
Theoremiccf1o 9345* Describe a bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵]. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) ∧ 𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦𝐴) / (𝐵𝐴)))))
 
Theoremunitssre 9346 (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(0[,]1) ⊆ ℝ
 
Theoremzltaddlt1le 9347 The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁))
 
3.5.4  Finite intervals of integers
 
Syntaxcfz 9348 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 ...
 
Definitiondf-fz 9349* 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 9350 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
 
Theoremfzval 9350* 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 11-2.1 of [Gleason] p. 141, where _k means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
 
Theoremfzval2 9351 An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
 
Theoremfzf 9352 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
...:(ℤ × ℤ)⟶𝒫 ℤ
 
Theoremelfz1 9353 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀𝐾𝐾𝑁)))
 
Theoremelfz 9354 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀𝐾𝐾𝑁)))
 
Theoremelfz2 9355 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) ∧ (𝑀𝐾𝐾𝑁)))
 
Theoremelfz5 9356 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾𝑁))
 
Theoremelfz4 9357 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) ∧ (𝑀𝐾𝐾𝑁)) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzuzb 9358 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 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)))
 
Theoremeluzfz 9359 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzuz 9360 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 ∈ (ℤ𝑀))
 
Theoremelfzuz3 9361 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 ∈ (ℤ𝐾))
 
Theoremelfzel2 9362 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)
 
Theoremelfzel1 9363 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)
 
Theoremelfzelz 9364 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)
 
Theoremelfzle1 9365 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) → 𝑀𝐾)
 
Theoremelfzle2 9366 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) → 𝐾𝑁)
 
Theoremelfzuz2 9367 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 ∈ (ℤ𝑀))
 
Theoremelfzle3 9368 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) → 𝑀𝑁)
 
Theoremeluzfz1 9369 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
 
Theoremeluzfz2 9370 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
 
Theoremeluzfz2b 9371 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
 
Theoremelfz3 9372 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
 
Theoremelfz1eq 9373 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
 
Theoremelfzubelfz 9374 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))
 
Theorempeano2fzr 9375 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
((𝐾 ∈ (ℤ𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremfzm 9376* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ𝑀))
 
Theoremfztri3or 9377 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))
 
Theoremfzdcel 9378 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))
 
Theoremfznlem 9379 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) = ∅))
 
Theoremfzn 9380 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) = ∅))
 
Theoremfzen 9381 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)))
 
Theoremfz1n 9382 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))
 
Theorem0fz1 9383 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0))
 
Theoremfz10 9384 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) = ∅
 
Theoremuzsubsubfz 9385 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 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐿)) → (𝑁 − (𝐿𝑀)) ∈ (𝑀...𝑁))
 
Theoremuzsubsubfz1 9386 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 ∈ (ℤ𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁))
 
Theoremige3m2fz 9387 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))
 
Theoremfzsplit2 9388 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(((𝐾 + 1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
 
Theoremfzsplit 9389 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)))
 
Theoremfzdisj 9390 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
 
Theoremfz01en 9391 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))
 
Theoremelfznn 9392 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)
 
Theoremelfz1end 9393 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
 
Theoremfznn0sub 9394 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) → (𝑁𝐾) ∈ ℕ0)
 
Theoremfzmmmeqm 9395 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) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
 
Theoremfzaddel 9396 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))))
 
Theoremfzsubel 9397 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) ↔ (𝐽𝐾) ∈ ((𝑀𝐾)...(𝑁𝐾))))
 
Theoremfzopth 9398 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 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
 
Theoremfzass4 9399 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)))
 
Theoremfzss1 9400 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (ℤ𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
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