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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrnepnf 8801 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrltnr 8802 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Theoremltpnf 8803 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theorem0ltpnf 8804 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnflt 8805 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

Theoremmnflt0 8806 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnfltpnf 8807 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltxr 8808 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Theorempnfnlt 8809 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremnltmnf 8810 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

Theorempnfge 8811 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)

Theorem0lepnf 8812 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremnn0pnfge0 8813 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 8814 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)

Theoremxrltnsym 8815 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Theoremxrltnsym2 8816 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremxrlttr 8817 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

Theoremxrltso 8818 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Theoremxrlttri3 8819 Extended real version of lttri3 7157. (Contributed by NM, 9-Feb-2006.)

Theoremxrltle 8820 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleid 8821 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri3 8822 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

Theoremxrlelttr 8823 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrltletr 8824 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrletr 8825 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrlttrd 8826 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrlelttrd 8827 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltletrd 8828 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrletrd 8829 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltne 8830 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Theoremnltpnft 8831 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Theoremngtmnft 8832 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Theoremxrrebnd 8833 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Theoremxrre 8834 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Theoremxrre2 8835 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Theoremxrre3 8836 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

Theoremge0gtmnf 8837 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremge0nemnf 8838 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrrege0 8839 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremz2ge 8840* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)

Theoremxnegeq 8841 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 8842 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Theoremxnegmnf 8843 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremrexneg 8844 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxneg0 8845 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcl 8846 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegneg 8847 Extended real version of negneg 7324. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxneg11 8848 Extended real version of neg11 7325. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltnegi 8849 Forward direction of xltneg 8850. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltneg 8850 Extended real version of ltneg 7531. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleneg 8851 Extended real version of leneg 7534. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg1 8852 Extended real version of lt0neg1 7537. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg2 8853 Extended real version of lt0neg2 7538. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxle0neg1 8854 Extended real version of le0neg1 7539. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxle0neg2 8855 Extended real version of le0neg2 7540. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxnegcld 8856 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxrex 8857 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)

3.5.3  Real number intervals

Syntaxcioo 8858 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 8859 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 8860 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 8861 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 8862* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 8863* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 8864* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 8865* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 8866* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 8867* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 8868* 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 8869* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 8870* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 8871* 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 8872* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 8873* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxss1 8874* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 8875* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 8876* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremiooex 8877 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooval 8878* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooidg 8879 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)

Theoremelioo3g 8880 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 8881 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioore 8882 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 8883 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremubioog 8884 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremiooval2 8885* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooss1 8886 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiooss2 8887 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiocval 8888* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremicoval 8889* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccval 8890* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo2 8891 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremelioc1 8892 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 8893 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 8894 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccid 8895 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremicc0r 8896 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)

Theoremeliooxr 8897 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooord 8898 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 8899 The upper bound belongs to an open-below, closed-above interval. See ubicc2 8954. (Contributed by FL, 29-May-2014.)

Theoremlbico1 8900 The lower bound belongs to a closed-below, open-above interval. See lbicc2 8953. (Contributed by FL, 29-May-2014.)

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