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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremioopos 9101 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,) +oo )  =  { x  e.  RR  |  0  < 
 x }
 
Theoremioorp 9102 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( 0 (,) +oo )  =  RR+
 
Theoremiooshf 9103 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  B )  e.  ( C (,) D )  <->  A  e.  (
 ( C  +  B ) (,) ( D  +  B ) ) ) )
 
Theoremiocssre 9104 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
 
Theoremicossre 9105 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( A [,) B )  C_  RR )
 
Theoremiccssre 9106 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  C_  RR )
 
Theoremiccssxr 9107 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  ( A [,] B )  C_  RR*
 
Theoremiocssxr 9108 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 )  C_  RR*
 
Theoremicossxr 9109 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 )  C_  RR*
 
Theoremioossicc 9110 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( A (,) B )  C_  ( A [,] B )
 
Theoremicossicc 9111 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 9112 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 9113 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 9114 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 9115 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B ) )  ->  ( C [,) D ) 
 C_  ( A (,) B ) )
 
Theoremioossioo 9116 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremiccsupr 9117* 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  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  y  <_  x ) )
 
Theoremelioopnf 9118 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  <  B ) ) )
 
Theoremelioomnf 9119 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\  B  <  A ) ) )
 
Theoremelicopnf 9120 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( A  e.  RR  ->  ( B  e.  ( A [,) +oo )  <->  ( B  e.  RR  /\  A  <_  B ) ) )
 
Theoremrepos 9121 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
 |-  ( A  e.  (
 0 (,) +oo )  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremioof 9122 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.)
 |- 
 (,) : ( RR*  X.  RR* )
 --> ~P RR
 
Theoremiccf 9123 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.)
 |- 
 [,] : ( RR*  X.  RR* )
 --> ~P RR*
 
Theoremunirnioo 9124 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
 |- 
 RR  =  U. ran  (,)
 
Theoremdfioo2 9125* 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  e.  RR* ,  y  e.  RR*  |->  { w  e.  RR  |  ( x  <  w  /\  w  <  y ) } )
 
Theoremioorebasg 9126 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ran  (,) )
 
Theoremelrege0 9127 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  (
 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrge0ssre 9128 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
 |-  ( 0 [,) +oo )  C_  RR
 
Theoremelxrge0 9129 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A  e.  (
 0 [,] +oo )  <->  ( A  e.  RR*  /\  0  <_  A ) )
 
Theorem0e0icopnf 9130 0 is a member of  ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,) +oo )
 
Theorem0e0iccpnf 9131 0 is a member of  ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,] +oo )
 
Theoremge0addcl 9132 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  +  B )  e.  ( 0 [,) +oo ) )
 
Theoremge0mulcl 9133 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  x.  B )  e.  ( 0 [,) +oo ) )
 
Theoremlbicc2 9134 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  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B ) )
 
Theoremubicc2 9135 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  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B ) )
 
Theorem0elunit 9136 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  0  e.  ( 0 [,] 1 )
 
Theorem1elunit 9137 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  1  e.  ( 0 [,] 1 )
 
Theoremiooneg 9138 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  -u C  e.  ( -u B (,) -u A ) ) )
 
Theoremiccneg 9139 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  -u C  e.  ( -u B [,] -u A ) ) )
 
Theoremicoshft 9140 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
 
Theoremicoshftf1o 9141* 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  e.  ( A [,) B )  |->  ( x  +  C ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  F :
 ( A [,) B )
 -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
 ) )
 
Theoremicodisj 9142 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A [,) B )  i^i  ( B [,) C ) )  =  (/) )
 
Theoremioodisj 9143 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  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  /\  B  <_  C )  ->  ( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
 
Theoremiccshftr 9144 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  +  R )  =  C   &    |-  ( B  +  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  +  R )  e.  ( C [,] D ) ) )
 
Theoremiccshftri 9145 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR   &    |-  ( A  +  R )  =  C   &    |-  ( B  +  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  +  R )  e.  ( C [,] D ) )
 
Theoremiccshftl 9146 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  -  R )  =  C   &    |-  ( B  -  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  -  R )  e.  ( C [,] D ) ) )
 
Theoremiccshftli 9147 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR   &    |-  ( A  -  R )  =  C   &    |-  ( B  -  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  -  R )  e.  ( C [,] D ) )
 
Theoremiccdil 9148 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  x.  R )  =  C   &    |-  ( B  x.  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )
 
Theoremiccdili 9149 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR+   &    |-  ( A  x.  R )  =  C   &    |-  ( B  x.  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  x.  R )  e.  ( C [,] D ) )
 
Theoremicccntr 9150 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  /  R )  =  C   &    |-  ( B  /  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
 
Theoremicccntri 9151 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR+   &    |-  ( A  /  R )  =  C   &    |-  ( B  /  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  /  R )  e.  ( C [,] D ) )
 
Theoremdivelunit 9152 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  (
 0 [,] 1 )  <->  A  <_  B ) )
 
Theoremlincmb01cmp 9153 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  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1
 ) )  ->  (
 ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  e.  ( A [,] B ) )
 
Theoremiccf1o 9154* Describe a bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) 
 /\  `' F  =  (
 y  e.  ( A [,] B )  |->  ( ( y  -  A )  /  ( B  -  A ) ) ) ) )
 
Theoremunitssre 9155  ( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( 0 [,] 1
 )  C_  RR
 
Theoremzltaddlt1le 9156 The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) ) 
 ->  ( ( M  +  A )  <  N  <->  ( M  +  A )  <_  N ) )
 
3.5.4  Finite intervals of integers
 
Syntaxcfz 9157 Extend class notation to include the notation for a contiguous finite set of integers. Read " M ... N " as "the set of integers from  M to  N inclusive."
 class  ...
 
Definitiondf-fz 9158* Define an operation that produces a finite set of sequential integers. Read " M ... N " as "the set of integers from  M to  N inclusive." See fzval 9159 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
 |- 
 ...  =  ( m  e.  ZZ ,  n  e. 
 ZZ  |->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) } )
 
Theoremfzval 9159* The value of a finite set of sequential integers. E.g.,  2 ... 5 means the set  { 2 ,  3 ,  4 ,  5 }. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where  NN_k means our  1 ... k; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  { k  e.  ZZ  |  ( M 
 <_  k  /\  k  <_  N ) } )
 
Theoremfzval2 9160 An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  ( ( M [,] N )  i^i  ZZ ) )
 
Theoremfzf 9161 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 ... : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfz1 9162 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
 ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
 
Theoremelfz 9163 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( M  <_  K 
 /\  K  <_  N ) ) )
 
Theoremelfz2 9164 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) ) )
 
Theoremelfz5 9165 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  K  <_  N ) )
 
Theoremelfz4 9166 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuzb 9167 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) ) )
 
Theoremeluzfz 9168 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuz 9169 A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M ) )
 
Theoremelfzuz3 9170 Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K ) )
 
Theoremelfzel2 9171 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
 
Theoremelfzel1 9172 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
 
Theoremelfzelz 9173 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
 
Theoremelfzle1 9174 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  K )
 
Theoremelfzle2 9175 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  <_  N )
 
Theoremelfzuz2 9176 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  M ) )
 
Theoremelfzle3 9177 Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  N )
 
Theoremeluzfz1 9178 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  e.  ( M ... N ) )
 
Theoremeluzfz2 9179 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  ( M ... N ) )
 
Theoremeluzfz2b 9180 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  <->  N  e.  ( M ... N ) )
 
Theoremelfz3 9181 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
 |-  ( N  e.  ZZ  ->  N  e.  ( N
 ... N ) )
 
Theoremelfz1eq 9182 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
 |-  ( K  e.  ( N ... N )  ->  K  =  N )
 
Theoremelfzubelfz 9183 If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( M ... N ) )
 
Theorempeano2fzr 9184 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M
 ... N ) ) 
 ->  K  e.  ( M
 ... N ) )
 
Theoremfzm 9185* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
 |-  ( E. x  x  e.  ( M ... N )  <->  N  e.  ( ZZ>=
 `  M ) )
 
Theoremfztri3or 9186 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  K  e.  ( M
 ... N )  \/  N  <  K ) )
 
Theoremfzdcel 9187 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  ( M ... N ) )
 
Theoremfznlem 9188 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M 
 ->  ( M ... N )  =  (/) ) )
 
Theoremfzn 9189 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <-> 
 ( M ... N )  =  (/) ) )
 
Theoremfzen 9190 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M ... N )  ~~  ( ( M  +  K ) ... ( N  +  K ) ) )
 
Theoremfz1n 9191 A 1-based finite set of sequential integers is empty iff it ends at index  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( ( 1 ...
 N )  =  (/)  <->  N  =  0 ) )
 
Theorem0fz1 9192 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( F  =  (/)  <->  N  =  0 ) )
 
Theoremfz10 9193 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( 1 ... 0
 )  =  (/)
 
Theoremuzsubsubfz 9194 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( ( L  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  L ) )  ->  ( N  -  ( L  -  M ) )  e.  ( M ... N ) )
 
Theoremuzsubsubfz1 9195 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( ( L  e.  NN  /\  N  e.  ( ZZ>=
 `  L ) ) 
 ->  ( N  -  ( L  -  1 ) )  e.  ( 1 ...
 N ) )
 
Theoremige3m2fz 9196 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  ( N  -  2
 )  e.  ( 1
 ... N ) )
 
Theoremfzsplit2 9197 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( ( K  +  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzsplit 9198 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzdisj 9199 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( K  <  M  ->  ( ( J ... K )  i^i  ( M
 ... N ) )  =  (/) )
 
Theoremfz01en 9200 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( N  e.  ZZ  ->  ( 0 ... ( N  -  1 ) ) 
 ~~  ( 1 ...
 N ) )
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