Theorem List for Intuitionistic Logic Explorer - 9101-9200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | ioopos 9101 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
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Theorem | ioorp 9102 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
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Theorem | iooshf 9103 |
Shift the arguments of the open interval function. (Contributed by NM,
17-Aug-2008.)
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Theorem | iocssre 9104 |
A closed-above interval with real upper bound is a set of reals.
(Contributed by FL, 29-May-2014.)
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     ![(,] (,]](_ioc.gif)    |
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Theorem | icossre 9105 |
A closed-below interval with real lower bound is a set of reals.
(Contributed by Mario Carneiro, 14-Jun-2014.)
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Theorem | iccssre 9106 |
A closed real interval is a set of reals. (Contributed by FL,
6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
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     ![[,] [,]](_icc.gif) 
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Theorem | iccssxr 9107 |
A closed interval is a set of extended reals. (Contributed by FL,
28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
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  ![[,] [,]](_icc.gif)   |
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Theorem | iocssxr 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.)
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  ![(,] (,]](_ioc.gif)   |
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Theorem | icossxr 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.)
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Theorem | ioossicc 9110 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
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      ![[,] [,]](_icc.gif)   |
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Theorem | icossicc 9111 |
A closed-below, open-above interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25-Oct-2016.)
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      ![[,] [,]](_icc.gif)   |
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Theorem | iocssicc 9112 |
A closed-above, open-below interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1-Apr-2017.)
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  ![(,] (,]](_ioc.gif)    ![[,] [,]](_icc.gif)   |
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Theorem | ioossico 9113 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
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Theorem | iocssioo 9114 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29-Mar-2017.)
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      ![(,] (,]](_ioc.gif)        |
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Theorem | icossioo 9115 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 29-Mar-2017.)
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Theorem | ioossioo 9116 |
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26-Sep-2017.)
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Theorem | iccsupr 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.)
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      ![[,] [,]](_icc.gif)    
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Theorem | elioopnf 9118 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elioomnf 9119 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elicopnf 9120 |
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16-Sep-2014.)
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Theorem | repos 9121 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
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Theorem | ioof 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.)
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Theorem | iccf 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.)
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![[,]
[,]](_icc.gif)        |
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Theorem | unirnioo 9124 |
The union of the range of the open interval function. (Contributed by
NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
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Theorem | dfioo2 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.)
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Theorem | ioorebasg 9126 |
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4-Apr-2020.)
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Theorem | elrege0 9127 |
The predicate "is a nonnegative real". (Contributed by Jeff Madsen,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
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Theorem | rge0ssre 9128 |
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
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Theorem | elxrge0 9129 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
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Theorem | 0e0icopnf 9130 |
0 is a member of   
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | 0e0iccpnf 9131 |
0 is a member of   
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | ge0addcl 9132 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
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Theorem | ge0mulcl 9133 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
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Theorem | lbicc2 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.)
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     ![[,] [,]](_icc.gif)    |
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Theorem | ubicc2 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.)
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     ![[,] [,]](_icc.gif)    |
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Theorem | 0elunit 9136 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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  ![[,] [,]](_icc.gif)   |
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Theorem | 1elunit 9137 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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  ![[,] [,]](_icc.gif)   |
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Theorem | iooneg 9138 |
Membership in a negated open real interval. (Contributed by Paul Chapman,
26-Nov-2007.)
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Theorem | iccneg 9139 |
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26-Nov-2007.)
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      ![[,] [,]](_icc.gif)      ![[,] [,]](_icc.gif)      |
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Theorem | icoshft 9140 |
A shifted real is a member of a shifted, closed-below, open-above real
interval. (Contributed by Paul Chapman, 25-Mar-2008.)
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Theorem | icoshftf1o 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.)
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Theorem | icodisj 9142 |
End-to-end closed-below, open-above real intervals are disjoint.
(Contributed by Mario Carneiro, 16-Jun-2014.)
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Theorem | ioodisj 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.)
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Theorem | iccshftr 9144 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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   ![[,] [,]](_icc.gif)   
  ![[,] [,]](_icc.gif)     |
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Theorem | iccshftri 9145 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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  ![[,] [,]](_icc.gif)  
   ![[,] [,]](_icc.gif)    |
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Theorem | iccshftl 9146 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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   ![[,] [,]](_icc.gif)   
  ![[,] [,]](_icc.gif)     |
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Theorem | iccshftli 9147 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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  ![[,] [,]](_icc.gif)      ![[,] [,]](_icc.gif)    |
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Theorem | iccdil 9148 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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   ![[,] [,]](_icc.gif)   
  ![[,] [,]](_icc.gif)     |
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Theorem | iccdili 9149 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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  ![[,] [,]](_icc.gif)      ![[,] [,]](_icc.gif)    |
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Theorem | icccntr 9150 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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   ![[,] [,]](_icc.gif)   
  ![[,] [,]](_icc.gif)     |
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Theorem | icccntri 9151 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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  ![[,] [,]](_icc.gif)      ![[,] [,]](_icc.gif)    |
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Theorem | divelunit 9152 |
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11-Jun-2013.)
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  ![[,] [,]](_icc.gif) 
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Theorem | lincmb01cmp 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.)
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      ![[,] [,]](_icc.gif)      

     ![[,] [,]](_icc.gif)    |
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Theorem | iccf1o 9154* |
Describe a bijection from    to an arbitrary nontrivial
closed interval    . (Contributed by Mario Carneiro,
8-Sep-2015.)
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   ![[,] [,]](_icc.gif)                   ![[,] [,]](_icc.gif)      ![[,] [,]](_icc.gif)  
   ![[,] [,]](_icc.gif)            |
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Theorem | unitssre 9155 |
  ![[,] [,]](_icc.gif)  is a subset of the reals.
(Contributed by David Moews,
28-Feb-2017.)
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  ![[,] [,]](_icc.gif) 
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Theorem | zltaddlt1le 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.)
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3.5.4 Finite intervals of integers
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Syntax | cfz 9157 |
Extend class notation to include the notation for a contiguous finite set
of integers. Read "  " as "the set of
integers from to
inclusive."
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Definition | df-fz 9158* |
Define an operation that produces a finite set of sequential integers.
Read "  " as "the set of integers from
to
inclusive." See fzval 9159 for its value and additional comments.
(Contributed by NM, 6-Sep-2005.)
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Theorem | fzval 9159* |
The value of a finite set of sequential integers. E.g.,  
means the set      . A special case of this definition
(starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where
_k means our
  ; he calls these sets segments of the
integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
3-Nov-2013.)
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Theorem | fzval2 9160 |
An alternate way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3-Nov-2013.)
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          ![[,] [,]](_icc.gif)     |
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Theorem | fzf 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.)
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Theorem | elfz1 9162 |
Membership in a finite set of sequential integers. (Contributed by NM,
21-Jul-2005.)
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Theorem | elfz 9163 |
Membership in a finite set of sequential integers. (Contributed by NM,
29-Sep-2005.)
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Theorem | elfz2 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
and . (Contributed by NM, 6-Sep-2005.)
(Revised by Mario
Carneiro, 28-Apr-2015.)
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Theorem | elfz5 9165 |
Membership in a finite set of sequential integers. (Contributed by NM,
26-Dec-2005.)
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Theorem | elfz4 9166 |
Membership in a finite set of sequential integers. (Contributed by NM,
21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | elfzuzb 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.)
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Theorem | eluzfz 9168 |
Membership in a finite set of sequential integers. (Contributed by NM,
4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | elfzuz 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.)
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Theorem | elfzuz3 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.)
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Theorem | elfzel2 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.)
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Theorem | elfzel1 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.)
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Theorem | elfzelz 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.)
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Theorem | elfzle1 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.)
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Theorem | elfzle2 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.)
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Theorem | elfzuz2 9176 |
Implication of membership in a finite set of sequential integers.
(Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | elfzle3 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.)
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Theorem | eluzfz1 9178 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | eluzfz2 9179 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro,
28-Apr-2015.)
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Theorem | eluzfz2b 9180 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 14-Sep-2005.)
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Theorem | elfz3 9181 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21-Jul-2005.)
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Theorem | elfz1eq 9182 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19-Sep-2005.)
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Theorem | elfzubelfz 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.)
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Theorem | peano2fzr 9184 |
A Peano-postulate-like theorem for downward closure of a finite set of
sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
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Theorem | fzm 9185* |
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15-Apr-2020.)
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Theorem | fztri3or 9186 |
Trichotomy in terms of a finite interval of integers. (Contributed by Jim
Kingdon, 1-Jun-2020.)
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Theorem | fzdcel 9187 |
Decidability of membership in a finite interval of integers. (Contributed
by Jim Kingdon, 1-Jun-2020.)
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   DECID       |
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Theorem | fznlem 9188 |
A finite set of sequential integers is empty if the bounds are reversed.
(Contributed by Jim Kingdon, 16-Apr-2020.)
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Theorem | fzn 9189 |
A finite set of sequential integers is empty if the bounds are reversed.
(Contributed by NM, 22-Aug-2005.)
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Theorem | fzen 9190 |
A shifted finite set of sequential integers is equinumerous to the
original set. (Contributed by Paul Chapman, 11-Apr-2009.)
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Theorem | fz1n 9191 |
A 1-based finite set of sequential integers is empty iff it ends at index
. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | 0fz1 9192 |
Two ways to say a finite 1-based sequence is empty. (Contributed by Paul
Chapman, 26-Oct-2012.)
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Theorem | fz10 9193 |
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
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Theorem | uzsubsubfz 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.)
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Theorem | uzsubsubfz1 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.)
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Theorem | ige3m2fz 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.)
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Theorem | fzsplit2 9197 |
Split a finite interval of integers into two parts. (Contributed by
Mario Carneiro, 13-Apr-2016.)
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Theorem | fzsplit 9198 |
Split a finite interval of integers into two parts. (Contributed by
Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
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Theorem | fzdisj 9199 |
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17-Jun-2010.)
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Theorem | fz01en 9200 |
0-based and 1-based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11-Apr-2009.)
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