Home Intuitionistic Logic ExplorerTheorem List (p. 88 of 102) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrltnr 8701 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Theoremltpnf 8702 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theorem0ltpnf 8703 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnflt 8704 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

Theoremmnflt0 8705 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnfltpnf 8706 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltxr 8707 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Theorempnfnlt 8708 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremnltmnf 8709 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

Theorempnfge 8710 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)

Theorem0lepnf 8711 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremnn0pnfge0 8712 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.)

Theoremmnfle 8713 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)

Theoremxrltnsym 8714 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Theoremxrltnsym2 8715 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremxrlttr 8716 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

Theoremxrltso 8717 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Theoremxrlttri3 8718 Extended real version of lttri3 7098. (Contributed by NM, 9-Feb-2006.)

Theoremxrltle 8719 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleid 8720 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri3 8721 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

Theoremxrlelttr 8722 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrltletr 8723 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrletr 8724 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrlttrd 8725 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrlelttrd 8726 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltletrd 8727 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrletrd 8728 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltne 8729 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Theoremnltpnft 8730 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Theoremngtmnft 8731 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Theoremxrrebnd 8732 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Theoremxrre 8733 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Theoremxrre2 8734 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Theoremxrre3 8735 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

Theoremge0gtmnf 8736 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremge0nemnf 8737 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrrege0 8738 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremz2ge 8739* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)

Theoremxnegeq 8740 Equality of two extended numbers with in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxnegpnf 8741 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Theoremxnegmnf 8742 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremrexneg 8743 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxneg0 8744 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcl 8745 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegneg 8746 Extended real version of negneg 7261. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxneg11 8747 Extended real version of neg11 7262. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltnegi 8748 Forward direction of xltneg 8749. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltneg 8749 Extended real version of ltneg 7457. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleneg 8750 Extended real version of leneg 7460. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg1 8751 Extended real version of lt0neg1 7463. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg2 8752 Extended real version of lt0neg2 7464. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxle0neg1 8753 Extended real version of le0neg1 7465. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxle0neg2 8754 Extended real version of le0neg2 7466. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxnegcld 8755 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxrex 8756 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)

3.5.3  Real number intervals

Syntaxcioo 8757 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 8758 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 8759 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 8760 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 8761* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 8762* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 8763* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 8764* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 8765* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 8766* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 8767* 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.)

Theoremixxex 8768* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 8769* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 8770* 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 and . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxssixx 8771* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 8772* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxss1 8773* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 8774* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 8775* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremiooex 8776 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooval 8777* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooidg 8778 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)

Theoremelioo3g 8779 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 and . (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1 8780 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioore 8781 A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremlbioog 8782 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremubioog 8783 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremiooval2 8784* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooss1 8785 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiooss2 8786 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiocval 8787* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremicoval 8788* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccval 8789* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo2 8790 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremelioc1 8791 Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelico1 8792 Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelicc1 8793 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccid 8794 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremicc0r 8795 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremeliooxr 8796 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooord 8797 Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremubioc1 8798 The upper bound belongs to an open-below, closed-above interval. See ubicc2 8853. (Contributed by FL, 29-May-2014.)

Theoremlbico1 8799 The lower bound belongs to a closed-below, open-above interval. See lbicc2 8852. (Contributed by FL, 29-May-2014.)

Theoremiccleub 8800 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
 Copyright terms: Public domain < Previous  Next >