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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxlt0neg1 8501 Extended real version of lt0neg1 7238. (Contributed by Mario Carneiro, 20-Aug-2015.)
(A * → (A < 0 ↔ 0 < -𝑒A))
 
Theoremxlt0neg2 8502 Extended real version of lt0neg2 7239. (Contributed by Mario Carneiro, 20-Aug-2015.)
(A * → (0 < A ↔ -𝑒A < 0))
 
Theoremxle0neg1 8503 Extended real version of le0neg1 7240. (Contributed by Mario Carneiro, 9-Sep-2015.)
(A * → (A ≤ 0 ↔ 0 ≤ -𝑒A))
 
Theoremxle0neg2 8504 Extended real version of le0neg2 7241. (Contributed by Mario Carneiro, 9-Sep-2015.)
(A * → (0 ≤ A ↔ -𝑒A ≤ 0))
 
Theoremxnegcld 8505 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA *)       (φ → -𝑒A *)
 
Theoremxrex 8506 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* V
 
3.5.3  Real number intervals
 
Syntaxcioo 8507 Extend class notation with the set of open intervals of extended reals.
class (,)
 
Syntaxcioc 8508 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
 
Syntaxcico 8509 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
 
Syntaxcicc 8510 Extend class notation with the set of closed intervals of extended reals.
class [,]
 
Definitiondf-ioo 8511* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (x *, y * ↦ {z * ∣ (x < z z < y)})
 
Definitiondf-ioc 8512* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,] = (x *, y * ↦ {z * ∣ (x < z zy)})
 
Definitiondf-ico 8513* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (x *, y * ↦ {z * ∣ (xz z < y)})
 
Definitiondf-icc 8514* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (x *, y * ↦ {z * ∣ (xz zy)})
 
Theoremixxval 8515* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       ((A * B *) → (A𝑂B) = {z * ∣ (A𝑅z z𝑆B)})
 
Theoremelixx1 8516* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       ((A * B *) → (𝐶 (A𝑂B) ↔ (𝐶 * A𝑅𝐶 𝐶𝑆B)))
 
Theoremixxf 8517* 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 *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremixxex 8518* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       𝑂 V
 
Theoremixxssxr 8519* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       (A𝑂B) ⊆ ℝ*
 
Theoremelixx3g 8520* 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 * and B *. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})       (𝐶 (A𝑂B) ↔ ((A * B * 𝐶 *) (A𝑅𝐶 𝐶𝑆B)))
 
Theoremixxssixx 8521* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((A * w *) → (A𝑅wA𝑇w))    &   ((w * B *) → (w𝑆Bw𝑈B))       (A𝑂B) ⊆ (A𝑃B)
 
Theoremixxdisj 8522* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((B * w *) → (B𝑇w ↔ ¬ w𝑆B))       ((A * B * 𝐶 *) → ((A𝑂B) ∩ (B𝑃𝐶)) = ∅)
 
Theoremixxss1 8523* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑆y)})    &   ((A * B * w *) → ((A𝑊B B𝑇w) → A𝑅w))       ((A * A𝑊B) → (B𝑃𝐶) ⊆ (A𝑂𝐶))
 
Theoremixxss2 8524* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑇y)})    &   ((w * B * 𝐶 *) → ((w𝑇B B𝑊𝐶) → w𝑆𝐶))       ((𝐶 * B𝑊𝐶) → (A𝑃B) ⊆ (A𝑂𝐶))
 
Theoremixxss12 8525* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((A * 𝐶 * w *) → ((A𝑊𝐶 𝐶𝑇w) → A𝑅w))    &   ((w * 𝐷 * B *) → ((w𝑈𝐷 𝐷𝑋B) → w𝑆B))       (((A * B *) (A𝑊𝐶 𝐷𝑋B)) → (𝐶𝑃𝐷) ⊆ (A𝑂B))
 
Theoremiooex 8526 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) V
 
Theoremiooval 8527* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,)B) = {x * ∣ (A < x x < B)})
 
Theoremiooidg 8528 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
(A * → (A(,)A) = ∅)
 
Theoremelioo3g 8529 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 * and B *. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐶 (A(,)B) ↔ ((A * B * 𝐶 *) (A < 𝐶 𝐶 < B)))
 
Theoremelioo1 8530 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A(,)B) ↔ (𝐶 * A < 𝐶 𝐶 < B)))
 
Theoremelioore 8531 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)
 
Theoremlbioog 8532 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → ¬ A (A(,)B))
 
Theoremubioog 8533 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → ¬ B (A(,)B))
 
Theoremiooval2 8534* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,)B) = {x ℝ ∣ (A < x x < B)})
 
Theoremiooss1 8535 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
((A * AB) → (B(,)𝐶) ⊆ (A(,)𝐶))
 
Theoremiooss2 8536 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐶 * B𝐶) → (A(,)B) ⊆ (A(,)𝐶))
 
Theoremiocval 8537* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,]B) = {x * ∣ (A < x xB)})
 
Theoremicoval 8538* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A[,)B) = {x * ∣ (Ax x < B)})
 
Theoremiccval 8539* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A[,]B) = {x * ∣ (Ax xB)})
 
Theoremelioo2 8540 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((A * B *) → (𝐶 (A(,)B) ↔ (𝐶 A < 𝐶 𝐶 < B)))
 
Theoremelioc1 8541 Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A(,]B) ↔ (𝐶 * A < 𝐶 𝐶B)))
 
Theoremelico1 8542 Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A[,)B) ↔ (𝐶 * A𝐶 𝐶 < B)))
 
Theoremelicc1 8543 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A[,]B) ↔ (𝐶 * A𝐶 𝐶B)))
 
Theoremiccid 8544 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
(A * → (A[,]A) = {A})
 
Theoremicc0r 8545 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → (B < A → (A[,]B) = ∅))
 
Theoremeliooxr 8546 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
(A (B(,)𝐶) → (B * 𝐶 *))
 
Theoremeliooord 8547 Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)
(A (B(,)𝐶) → (B < A A < 𝐶))
 
Theoremubioc1 8548 The upper bound belongs to an open-below, closed-above interval. See ubicc2 8603. (Contributed by FL, 29-May-2014.)
((A * B * A < B) → B (A(,]B))
 
Theoremlbico1 8549 The lower bound belongs to a closed-below, open-above interval. See lbicc2 8602. (Contributed by FL, 29-May-2014.)
((A * B * A < B) → A (A[,)B))
 
Theoremiccleub 8550 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
((A * B * 𝐶 (A[,]B)) → 𝐶B)
 
Theoremiccgelb 8551 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
((A * B * 𝐶 (A[,]B)) → A𝐶)
 
Theoremelioo5 8552 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
((A * B * 𝐶 *) → (𝐶 (A(,)B) ↔ (A < 𝐶 𝐶 < B)))
 
Theoremelioo4g 8553 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐶 (A(,)B) ↔ ((A * B * 𝐶 ℝ) (A < 𝐶 𝐶 < B)))
 
Theoremioossre 8554 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(A(,)B) ⊆ ℝ
 
Theoremelioc2 8555 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A * B ℝ) → (𝐶 (A(,]B) ↔ (𝐶 A < 𝐶 𝐶B)))
 
Theoremelico2 8556 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A B *) → (𝐶 (A[,)B) ↔ (𝐶 A𝐶 𝐶 < B)))
 
Theoremelicc2 8557 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A B ℝ) → (𝐶 (A[,]B) ↔ (𝐶 A𝐶 𝐶B)))
 
Theoremelicc2i 8558 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
A     &   B        (𝐶 (A[,]B) ↔ (𝐶 A𝐶 𝐶B))
 
Theoremelicc4 8559 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
((A * B * 𝐶 *) → (𝐶 (A[,]B) ↔ (A𝐶 𝐶B)))
 
Theoremiccss 8560 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
(((A B ℝ) (A𝐶 𝐷B)) → (𝐶[,]𝐷) ⊆ (A[,]B))
 
Theoremiccssioo 8561 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
(((A * B *) (A < 𝐶 𝐷 < B)) → (𝐶[,]𝐷) ⊆ (A(,)B))
 
Theoremicossico 8562 Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
(((A * B *) (A𝐶 𝐷B)) → (𝐶[,)𝐷) ⊆ (A[,)B))
 
Theoremiccss2 8563 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐶 (A[,]B) 𝐷 (A[,]B)) → (𝐶[,]𝐷) ⊆ (A[,]B))
 
Theoremiccssico 8564 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((A * B *) (A𝐶 𝐷 < B)) → (𝐶[,]𝐷) ⊆ (A[,)B))
 
Theoremiccssioo2 8565 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐶 (A(,)B) 𝐷 (A(,)B)) → (𝐶[,]𝐷) ⊆ (A(,)B))
 
Theoremiccssico2 8566 Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐶 (A[,)B) 𝐷 (A[,)B)) → (𝐶[,]𝐷) ⊆ (A[,)B))
 
Theoremioomax 8567 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
(-∞(,)+∞) = ℝ
 
Theoremiccmax 8568 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
(-∞[,]+∞) = ℝ*
 
Theoremioopos 8569 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
(0(,)+∞) = {x ℝ ∣ 0 < x}
 
Theoremioorp 8570 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(0(,)+∞) = ℝ+
 
Theoremiooshf 8571 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((AB) (𝐶(,)𝐷) ↔ A ((𝐶 + B)(,)(𝐷 + B))))
 
Theoremiocssre 8572 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
((A * B ℝ) → (A(,]B) ⊆ ℝ)
 
Theoremicossre 8573 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
((A B *) → (A[,)B) ⊆ ℝ)
 
Theoremiccssre 8574 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
((A B ℝ) → (A[,]B) ⊆ ℝ)
 
Theoremiccssxr 8575 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
(A[,]B) ⊆ ℝ*
 
Theoremiocssxr 8576 An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
(A(,]B) ⊆ ℝ*
 
Theoremicossxr 8577 A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
(A[,)B) ⊆ ℝ*
 
Theoremioossicc 8578 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
(A(,)B) ⊆ (A[,]B)
 
Theoremicossicc 8579 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
(A[,)B) ⊆ (A[,]B)
 
Theoremiocssicc 8580 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
(A(,]B) ⊆ (A[,]B)
 
Theoremioossico 8581 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
(A(,)B) ⊆ (A[,)B)
 
Theoremiocssioo 8582 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((A * B *) (A𝐶 𝐷 < B)) → (𝐶(,]𝐷) ⊆ (A(,)B))
 
Theoremicossioo 8583 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((A * B *) (A < 𝐶 𝐷B)) → (𝐶[,)𝐷) ⊆ (A(,)B))
 
Theoremioossioo 8584 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((A * B *) (A𝐶 𝐷B)) → (𝐶(,)𝐷) ⊆ (A(,)B))
 
Theoremiccsupr 8585* 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 B ℝ) 𝑆 ⊆ (A[,]B) 𝐶 𝑆) → (𝑆 ⊆ ℝ 𝑆 ≠ ∅ x y 𝑆 yx))
 
Theoremelioopnf 8586 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(A * → (B (A(,)+∞) ↔ (B A < B)))
 
Theoremelioomnf 8587 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(A * → (B (-∞(,)A) ↔ (B B < A)))
 
Theoremelicopnf 8588 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
(A ℝ → (B (A[,)+∞) ↔ (B AB)))
 
Theoremrepos 8589 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
(A (0(,)+∞) ↔ (A 0 < A))
 
Theoremioof 8590 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 8591 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 8592 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 8593* 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 *, y * ↦ {w ℝ ∣ (x < w w < y)})
 
Theoremioorebasg 8594 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
((A * B *) → (A(,)B) ran (,))
 
Theoremelrege0 8595 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(A (0[,)+∞) ↔ (A 0 ≤ A))
 
Theoremrge0ssre 8596 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
(0[,)+∞) ⊆ ℝ
 
Theoremelxrge0 8597 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(A (0[,]+∞) ↔ (A * 0 ≤ A))
 
Theorem0e0icopnf 8598 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 (0[,)+∞)
 
Theorem0e0iccpnf 8599 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 (0[,]+∞)
 
Theoremge0addcl 8600 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
((A (0[,)+∞) B (0[,)+∞)) → (A + B) (0[,)+∞))
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