P. 134 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eliooxr 13301	A nonempty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*))
Theorem	eliooord 13302	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))
Theorem	elioo4g 13303	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 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	ioossre 13304	An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
⊢ (𝐴(,)𝐵) ⊆ ℝ
Theorem	ioosscn 13305	An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴(,)𝐵) ⊆ ℂ
Theorem	elioc2 13306	Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	elico2 13307	Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
Theorem	elicc2 13308	Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	elicc2i 13309	Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
Theorem	elicc4 13310	Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
Theorem	iccss 13311	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 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))
Theorem	iccssioo 13312	Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
Theorem	icossico 13313	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 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴[,)𝐵))
Theorem	iccss2 13314	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))
Theorem	iccssico 13315	Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
Theorem	iccssioo2 13316	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))
Theorem	iccssico2 13317	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))
Theorem	icossico2d 13318	Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))
Theorem	ioomax 13319	The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
⊢ (-∞(,)+∞) = ℝ
Theorem	iccmax 13320	The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
⊢ (-∞[,]+∞) = ℝ*
Theorem	ioopos 13321	The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
Theorem	ioorp 13322	The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (0(,)+∞) = ℝ+
Theorem	iooshf 13323	Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐵) ∈ (𝐶(,)𝐷) ↔ 𝐴 ∈ ((𝐶 + 𝐵)(,)(𝐷 + 𝐵))))
Theorem	iocssre 13324	A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
Theorem	icossre 13325	A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ)
Theorem	iccssre 13326	A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
Theorem	iccssxr 13327	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*
Theorem	iocssxr 13328	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*
Theorem	icossxr 13329	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*
Theorem	ioossicc 13330	An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
Theorem	iccssred 13331	A closed real interval is a set of reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
Theorem	eliccxr 13332	A member of a closed interval is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*)
Theorem	icossicc 13333	A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)
Theorem	iocssicc 13334	A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
Theorem	ioossico 13335	An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
Theorem	iocssioo 13336	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 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶(,]𝐷) ⊆ (𝐴(,)𝐵))
Theorem	icossioo 13337	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 < 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴(,)𝐵))
Theorem	ioossioo 13338	Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
Theorem	iccsupr 13339*	A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 12079). (Contributed by Paul Chapman, 21-Jan-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
Theorem	elioopnf 13340	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))
Theorem	elioomnf 13341	Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))
Theorem	elicopnf 13342	Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵)))
Theorem	repos 13343	Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
⊢ (𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	ioof 13344	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
Theorem	iccf 13345	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*
Theorem	unirnioo 13346	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 (,)
Theorem	dfioo2 13347*	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 < 𝑤 ∧ 𝑤 < 𝑦)})
Theorem	ioorebas 13348	Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)
⊢ (𝐴(,)𝐵) ∈ ran (,)
Theorem	xrge0neqmnf 13349	A nonnegative extended real is not equal to minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.) (Proof shortened by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)
Theorem	xrge0nre 13350	An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 = +∞)
Theorem	elrege0 13351	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))
Theorem	nn0rp0 13352	A nonnegative integer is a nonnegative real number. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0[,)+∞))
Theorem	rge0ssre 13353	Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
⊢ (0[,)+∞) ⊆ ℝ
Theorem	elxrge0 13354	Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴))
Theorem	0e0icopnf 13355	0 is a member of (0[,)+∞). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ∈ (0[,)+∞)
Theorem	0e0iccpnf 13356	0 is a member of (0[,]+∞). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ∈ (0[,]+∞)
Theorem	ge0addcl 13357	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 + 𝐵) ∈ (0[,)+∞))
Theorem	ge0mulcl 13358	The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 · 𝐵) ∈ (0[,)+∞))
Theorem	ge0xaddcl 13359	The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))
Theorem	ge0xmulcl 13360	The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 ·e 𝐵) ∈ (0[,]+∞))
Theorem	lbicc2 13361	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* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
Theorem	ubicc2 13362	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* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
Theorem	elicc01 13363	Membership in the closed real interval between 0 and 1, also called the closed unit interval. (Contributed by AV, 20-Aug-2022.)
⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1))
Theorem	elunitrn 13364	The closed unit interval is a subset of the set of the real numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 𝐴 ∈ ℝ)
Theorem	elunitcn 13365	The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 𝐴 ∈ ℂ)
Theorem	0elunit 13366	Zero is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ 0 ∈ (0[,]1)
Theorem	1elunit 13367	One is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ 1 ∈ (0[,]1)
Theorem	iooneg 13368	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)))
Theorem	iccneg 13369	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)))
Theorem	icoshft 13370	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))))
Theorem	icoshftf1o 13371*	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)))
Theorem	icoun 13372	The union of two adjacent left-closed right-open real intervals is a left-closed right-open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))
Theorem	icodisj 13373	Adjacent left-closed right-open real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
Theorem	ioounsn 13374	The union of an open interval with its upper endpoint is a left-open right-closed interval. (Contributed by Jon Pennant, 8-Jun-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
Theorem	snunioo 13375	The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
Theorem	snunico 13376	The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
Theorem	snunioc 13377	The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
Theorem	prunioo 13378	The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
Theorem	ioodisj 13379	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*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅)
Theorem	ioojoin 13380	Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
Theorem	difreicc 13381	The class difference of ℝ and a closed interval. (Contributed by FL, 18-Jun-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞)))
Theorem	iccsplit 13382	Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝐴[,]𝐶) ∪ (𝐶[,]𝐵)))
Theorem	iccshftr 13383	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)))
Theorem	iccshftri 13384	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))
Theorem	iccshftl 13385	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)))
Theorem	iccshftli 13386	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))
Theorem	iccdil 13387	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)))
Theorem	iccdili 13388	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))
Theorem	icccntr 13389	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)))
Theorem	icccntri 13390	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))
Theorem	divelunit 13391	A condition for a ratio to be a member of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴 ≤ 𝐵))
Theorem	lincmb01cmp 13392	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))
Theorem	iccf1o 13393*	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) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴)))))
Theorem	iccen 13394	Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ≈ (𝐴[,]𝐵))
Theorem	xov1plusxeqvd 13395	A complex number 𝑋 is positive real iff 𝑋 / (1 + 𝑋) is in (0(,)1). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ -1)    ⇒   ⊢ (𝜑 → (𝑋 ∈ ℝ+ ↔ (𝑋 / (1 + 𝑋)) ∈ (0(,)1)))
Theorem	unitssre 13396	(0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
⊢ (0[,]1) ⊆ ℝ
Theorem	unitsscn 13397	The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ (0[,]1) ⊆ ℂ
Theorem	supicc 13398	Supremum of a bounded set of real numbers. (Contributed by Thierry Arnoux, 17-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶))
Theorem	supiccub 13399	The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐷 ≤ sup(𝐴, ℝ, < ))
Theorem	supicclub 13400*	The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧))