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Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-i2m1 10001 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 9977. (Contributed by NM, 29-Jan-1995.)
((i · i) + 1) = 0

Axiomax-1ne0 10002 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 9978. (Contributed by NM, 29-Jan-1995.)
1 ≠ 0

Axiomax-1rid 10003 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 9979. Weakened from the original axiom in the form of statement in mulid1 10034, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)

Axiomax-rnegex 10004* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 9980. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Axiomax-rrecex 10005* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9981. (Contributed by Eric Schmidt, 11-Apr-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

Axiomax-cnre 10006* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 9982. For naming consistency, use cnre 10033 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Axiomax-pre-lttri 10007 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9983. Note: The more general version for extended reals is axlttri 10106. Normally new proofs would use xrlttri 11969. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Axiomax-pre-lttrn 10008 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9984. Note: The more general version for extended reals is axlttrn 10107. Normally new proofs would use lttr 10111. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Axiomax-pre-ltadd 10009 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9985. Normally new proofs would use axltadd 10108. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Axiomax-pre-mulgt0 10010 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9986. Normally new proofs would use axmulgt0 10109. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Axiomax-pre-sup 10011* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 9987. Note: Normally new proofs would use axsup 10110. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Axiomax-addf 10012 Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 10015 should be used. Note that uses of ax-addf 10012 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) in place of +, from which this axiom (with the defined operation in place of +) follows as a theorem.

This axiom is justified by theorem axaddf 9963. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

+ :(ℂ × ℂ)⟶ℂ

Axiomax-mulf 10013 Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 9995 should be used. Note that uses of ax-mulf 10013 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, from which this axiom (with the defined operation in place of ·) follows as a theorem.

This axiom is justified by theorem axmulf 9964. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

· :(ℂ × ℂ)⟶ℂ

5.2  Derive the basic properties from the field axioms

5.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 10014 Alias for ax-cnex 9989. See also cnexALT 11825. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℂ ∈ V

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)

Theoremmulcl 10017 Alias for ax-mulcl 9995, for naming consistency with mulcli 10042. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Theoremremulcl 10018 Alias for ax-mulrcl 9996, for naming consistency with remulcli 10051. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)

Theoremmulcom 10019 Alias for ax-mulcom 9997, for naming consistency with mulcomi 10043. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulass 10021 Alias for ax-mulass 9999, for naming consistency with mulassi 10046. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddi 10022 Alias for ax-distr 10000, for naming consistency with adddii 10047. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremrecn 10023 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℂ)

Theoremreex 10024 The real numbers form a set. See also reexALT 11823. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V

Theoremreelprrecn 10025 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
ℝ ∈ {ℝ, ℂ}

Theoremcnelprrecn 10026 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
ℂ ∈ {ℝ, ℂ}

Theoremelimne0 10027 Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Theoremadddir 10028 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theorem0cn 10029 0 is a complex number. See also 0cnALT 10267. (Contributed by NM, 19-Feb-2005.)
0 ∈ ℂ

Theorem0cnd 10030 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℂ)

Theoremc0ex 10031 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V

Theorem1ex 10032 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V

Theoremcnre 10033* Alias for ax-cnre 10006, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Theoremmulid1 10034 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)

Theoremmulid2 10035 Identity law for multiplication. Note: see mulid1 10034 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)

Theorem1re 10036 1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 9991, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
1 ∈ ℝ

Theorem0re 10037 0 is a real number. See also 0reALT 10375. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0 ∈ ℝ

Theorem0red 10038 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 0 ∈ ℝ)

Theoremmulid1i 10039 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (𝐴 · 1) = 𝐴

Theoremmulid2i 10040 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (1 · 𝐴) = 𝐴

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) ∈ ℂ

Theoremmulcli 10042 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) ∈ ℂ

Theoremmulcomi 10043 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) = (𝐵 · 𝐴)

Theoremmulcomli 10044 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 · 𝐵) = 𝐶       (𝐵 · 𝐴) = 𝐶

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Theoremmulassi 10046 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Theoremadddii 10047 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Theoremadddiri 10048 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))

Theoremrecni 10049 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
𝐴 ∈ ℝ       𝐴 ∈ ℂ

Theoremreaddcli 10050 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 + 𝐵) ∈ ℝ

Theoremremulcli 10051 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 · 𝐵) ∈ ℝ

Theorem1red 10052 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)

Theorem1cnd 10053 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)

Theoremmulid1d 10054 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)

Theoremmulid2d 10055 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1 · 𝐴) = 𝐴)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) ∈ ℂ)

Theoremmulcld 10057 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)

Theoremmulcomd 10058 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulassd 10060 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddid 10061 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremadddird 10062 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremadddirp1d 10063 Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))

Theoremjoinlmuladdmuld 10064 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Theoremrecnd 10065 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℂ)

Theoremreaddcld 10066 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ)

Theoremremulcld 10067 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ)

5.2.2  Infinity and the extended real number system

Syntaxcpnf 10068 Plus infinity.
class +∞

Syntaxcmnf 10069 Minus infinity.
class -∞

Syntaxcxr 10070 The set of extended reals (includes plus and minus infinity).
class *

Syntaxclt 10071 'Less than' predicate (extended to include the extended reals).
class <

Syntaxcle 10072 Extend wff notation to include the 'less than or equal to' relation.
class

Definitiondf-pnf 10073 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 10074). We use 𝒫 to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 10078, mnfnre 10079, and pnfnemnf 10091.

A simpler possibility is to define +∞ as and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

+∞ = 𝒫

Definitiondf-mnf 10074 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in and different from +∞ (see mnfnre 10079 and pnfnemnf 10091). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞

Definitiondf-xr 10075 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)
* = (ℝ ∪ {+∞, -∞})

Definitiondf-ltxr 10076* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, < is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)
< = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))

Definitiondf-le 10077 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 10121 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )

Theorempnfnre 10078 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ

Theoremmnfnre 10079 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ

Theoremressxr 10080 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*

Theoremrexpssxrxp 10081 The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Theoremrexr 10082 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)

Theorem0xr 10083 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 ∈ ℝ*

Theoremrenepnf 10084 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ +∞)

Theoremrenemnf 10085 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ -∞)

Theoremrexrd 10086 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ*)

Theoremrenepnfd 10087 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ +∞)

Theoremrenemnfd 10088 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ -∞)

Theorempnfxr 10089 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+∞ ∈ ℝ*

Theorempnfex 10090 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ ∈ V

Theorempnfnemnf 10091 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞

Theoremmnfnepnf 10092 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞

Theoremmnfxr 10093 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
-∞ ∈ ℝ*

Theoremrexri 10094 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       𝐴 ∈ ℝ*

Theoremrenfdisj 10095 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅

Theoremltrelxr 10096 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)

Theoremltrel 10097 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <

Theoremlerelxr 10098 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)

Theoremlerel 10099 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤

Theoremxrlenlt 10100 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

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