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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltdiv1dd 8901 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 / 𝐶) < (𝐵 / 𝐶))
 
Theoremlediv1dd 8902 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))
 
Theoremlediv12ad 8903 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))
 
Theoremltdiv23d 8904 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 / 𝐵) < 𝐶)       (𝜑 → (𝐴 / 𝐶) < 𝐵)
 
Theoremlediv23d 8905 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 / 𝐵) ≤ 𝐶)       (𝜑 → (𝐴 / 𝐶) ≤ 𝐵)
 
Theoremlt2mul2divd 8906 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵)))
 
Theoremnnledivrp 8907 Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴))
 
Theoremnn0ledivnn 8908 Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴)
 
Theoremaddlelt 8909 If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)
((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁𝑀 < 𝑁))
 
3.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 8910 Extend class notation to include the negative of an extended real.
class -𝑒𝐴
 
Syntaxcxad 8911 Extend class notation to include addition of extended reals.
class +𝑒
 
Syntaxcxmu 8912 Extend class notation to include multiplication of extended reals.
class ·e
 
Definitiondf-xneg 8913 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
-𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
 
Definitiondf-xadd 8914* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
+𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
 
Definitiondf-xmul 8915* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
 
Theoremltxr 8916 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
 
Theoremelxr 8917 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
 
Theoremxrnemnf 8918 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
 
Theoremxrnepnf 8919 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
 
Theoremxrltnr 8920 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
 
Theoremltpnf 8921 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 < +∞)
 
Theorem0ltpnf 8922 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 < +∞
 
Theoremmnflt 8923 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → -∞ < 𝐴)
 
Theoremmnflt0 8924 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ < 0
 
Theoremmnfltpnf 8925 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-∞ < +∞
 
Theoremmnfltxr 8926 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
 
Theorempnfnlt 8927 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
 
Theoremnltmnf 8928 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
 
Theorempnfge 8929 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ*𝐴 ≤ +∞)
 
Theorem0lepnf 8930 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞
 
Theoremnn0pnfge0 8931 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑁 ∈ ℕ0𝑁 = +∞) → 0 ≤ 𝑁)
 
Theoremmnfle 8932 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(𝐴 ∈ ℝ* → -∞ ≤ 𝐴)
 
Theoremxrltnsym 8933 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
 
Theoremxrltnsym2 8934 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
 
Theoremxrlttr 8935 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremxrltso 8936 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*
 
Theoremxrlttri3 8937 Extended real version of lttri3 7247. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremxrltle 8938 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵𝐴𝐵))
 
Theoremxrleid 8939 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(𝐴 ∈ ℝ*𝐴𝐴)
 
Theoremxrletri3 8940 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremxrlelttr 8941 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremxrltletr 8942 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))
 
Theoremxrletr 8943 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremxrlttrd 8944 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrlelttrd 8945 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrltletrd 8946 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴 < 𝐶)
 
Theoremxrletrd 8947 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremxrltne 8948 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵𝐴)
 
Theoremnltpnft 8949 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
 
Theoremngtmnft 8950 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
 
Theoremxrrebnd 8951 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴𝐴 < +∞)))
 
Theoremxrre 8952 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (-∞ < 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)
 
Theoremxrre2 8953 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
 
Theoremxrre3 8954 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (𝐵𝐴𝐴 < +∞)) → 𝐴 ∈ ℝ)
 
Theoremge0gtmnf 8955 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
 
Theoremge0nemnf 8956 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
 
Theoremxrrege0 8957 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)
 
Theoremz2ge 8958* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀𝑘𝑁𝑘))
 
Theoremxnegeq 8959 Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
 
Theoremxnegpnf 8960 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞
 
Theoremxnegmnf 8961 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞
 
Theoremrexneg 8962 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
 
Theoremxneg0 8963 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0
 
Theoremxnegcl 8964 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
 
Theoremxnegneg 8965 Extended real version of negneg 7414. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
 
Theoremxneg11 8966 Extended real version of neg11 7415. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))
 
Theoremxltnegi 8967 Forward direction of xltneg 8968. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
 
Theoremxltneg 8968 Extended real version of ltneg 7622. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
 
Theoremxleneg 8969 Extended real version of leneg 7625. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
 
Theoremxlt0neg1 8970 Extended real version of lt0neg1 7628. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
 
Theoremxlt0neg2 8971 Extended real version of lt0neg2 7629. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
 
Theoremxle0neg1 8972 Extended real version of le0neg1 7630. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
 
Theoremxle0neg2 8973 Extended real version of le0neg2 7631. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
 
Theoremxnegcld 8974 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)
 
Theoremxrex 8975 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
3.5.3  Real number intervals
 
Syntaxcioo 8976 Extend class notation with the set of open intervals of extended reals.
class (,)
 
Syntaxcioc 8977 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
 
Syntaxcico 8978 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
 
Syntaxcicc 8979 Extend class notation with the set of closed intervals of extended reals.
class [,]
 
Definitiondf-ioo 8980* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
 
Definitiondf-ioc 8981* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧𝑦)})
 
Definitiondf-ico 8982* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
 
Definitiondf-icc 8983* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
 
Theoremixxval 8984* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})
 
Theoremelixx1 8985* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵)))
 
Theoremixxf 8986* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremixxex 8987* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂 ∈ V
 
Theoremixxssxr 8988* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐴𝑂𝐵) ⊆ ℝ*
 
Theoremelixx3g 8989* Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝑅𝐶𝐶𝑆𝐵)))
 
Theoremixxssixx 8990* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))       (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
 
Theoremixxdisj 8991* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)
 
Theoremixxss1 8992* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵𝐵𝑇𝑤) → 𝐴𝑅𝑤))       ((𝐴 ∈ ℝ*𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))
 
Theoremixxss2 8993* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑇𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵𝐵𝑊𝐶) → 𝑤𝑆𝐶))       ((𝐶 ∈ ℝ*𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))
 
Theoremixxss12 8994* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐶𝐶𝑇𝑤) → 𝐴𝑅𝑤))    &   ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝑈𝐷𝐷𝑋𝐵) → 𝑤𝑆𝐵))       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑊𝐶𝐷𝑋𝐵)) → (𝐶𝑃𝐷) ⊆ (𝐴𝑂𝐵))
 
Theoremiooex 8995 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) ∈ V
 
Theoremiooval 8996* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥 < 𝐵)})
 
Theoremiooidg 8997 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
(𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅)
 
Theoremelioo3g 8998 Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐶 < 𝐵)))
 
Theoremelioo1 8999 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶 < 𝐵)))
 
Theoremelioore 9000 A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)
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