Theorem List for Intuitionistic Logic Explorer - 8801-8900 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | iccgelb 8801 |
An element of a closed interval is more than or equal to its lower bound
(Contributed by Thierry Arnoux, 23-Dec-2016.)
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Theorem | elioo5 8802 |
Membership in an open interval of extended reals. (Contributed by NM,
17-Aug-2008.)
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Theorem | elioo4g 8803 |
Membership in an open interval of extended reals. (Contributed by NM,
8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | ioossre 8804 |
An open interval is a set of reals. (Contributed by NM,
31-May-2007.)
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Theorem | elioc2 8805 |
Membership in an open-below, closed-above real interval. (Contributed
by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro,
14-Jun-2014.)
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Theorem | elico2 8806 |
Membership in a closed-below, open-above real interval. (Contributed by
Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro,
14-Jun-2014.)
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Theorem | elicc2 8807 |
Membership in a closed real interval. (Contributed by Paul Chapman,
21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
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Theorem | elicc2i 8808 |
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3-Jun-2013.)
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Theorem | elicc4 8809 |
Membership in a closed real interval. (Contributed by Stefan O'Rear,
16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
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Theorem | iccss 8810 |
Condition for a closed interval to be a subset of another closed
interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario
Carneiro, 20-Feb-2015.)
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Theorem | iccssioo 8811 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20-Feb-2015.)
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Theorem | icossico 8812 |
Condition for a closed-below, open-above interval to be a subset of a
closed-below, open-above interval. (Contributed by Thierry Arnoux,
21-Sep-2017.)
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Theorem | iccss2 8813 |
Condition for a closed interval to be a subset of another closed
interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario
Carneiro, 28-Apr-2015.)
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Theorem | iccssico 8814 |
Condition for a closed interval to be a subset of a half-open interval.
(Contributed by Mario Carneiro, 9-Sep-2015.)
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Theorem | iccssioo2 8815 |
Condition for a closed interval to be a subset of an open interval.
(Contributed by Mario Carneiro, 20-Feb-2015.)
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Theorem | iccssico2 8816 |
Condition for a closed interval to be a subset of a closed-below,
open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
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Theorem | ioomax 8817 |
The open interval from minus to plus infinity. (Contributed by NM,
6-Feb-2007.)
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Theorem | iccmax 8818 |
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4-Jul-2014.)
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Theorem | ioopos 8819 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
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Theorem | ioorp 8820 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
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Theorem | iooshf 8821 |
Shift the arguments of the open interval function. (Contributed by NM,
17-Aug-2008.)
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Theorem | iocssre 8822 |
A closed-above interval with real upper bound is a set of reals.
(Contributed by FL, 29-May-2014.)
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Theorem | icossre 8823 |
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 8824 |
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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Theorem | iccssxr 8825 |
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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Theorem | iocssxr 8826 |
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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Theorem | icossxr 8827 |
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 8828 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
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Theorem | icossicc 8829 |
A closed-below, open-above interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25-Oct-2016.)
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Theorem | iocssicc 8830 |
A closed-above, open-below interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1-Apr-2017.)
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Theorem | ioossico 8831 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
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Theorem | iocssioo 8832 |
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 | icossioo 8833 |
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 8834 |
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 8835* |
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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Theorem | elioopnf 8836 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elioomnf 8837 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elicopnf 8838 |
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16-Sep-2014.)
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Theorem | repos 8839 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
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Theorem | ioof 8840 |
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 8841 |
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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Theorem | unirnioo 8842 |
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 8843* |
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 8844 |
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4-Apr-2020.)
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Theorem | elrege0 8845 |
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 8846 |
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 8847 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
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Theorem | 0e0icopnf 8848 |
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | 0e0iccpnf 8849 |
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | ge0addcl 8850 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
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Theorem | ge0mulcl 8851 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
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Theorem | lbicc2 8852 |
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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Theorem | ubicc2 8853 |
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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Theorem | 0elunit 8854 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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Theorem | 1elunit 8855 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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Theorem | iooneg 8856 |
Membership in a negated open real interval. (Contributed by Paul Chapman,
26-Nov-2007.)
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Theorem | iccneg 8857 |
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26-Nov-2007.)
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Theorem | icoshft 8858 |
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 8859* |
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 8860 |
End-to-end closed-below, open-above real intervals are disjoint.
(Contributed by Mario Carneiro, 16-Jun-2014.)
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Theorem | ioodisj 8861 |
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 8862 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftri 8863 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftl 8864 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftli 8865 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccdil 8866 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccdili 8867 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | icccntr 8868 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | icccntri 8869 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | divelunit 8870 |
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11-Jun-2013.)
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Theorem | lincmb01cmp 8871 |
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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Theorem | iccf1o 8872* |
Describe a bijection from to an arbitrary nontrivial
closed interval . (Contributed by Mario Carneiro,
8-Sep-2015.)
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Theorem | unitssre 8873 |
is a subset of the reals.
(Contributed by David Moews,
28-Feb-2017.)
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3.5.4 Finite intervals of integers
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Syntax | cfz 8874 |
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 8875* |
Define an operation that produces a finite set of sequential integers.
Read " " as "the set of integers from
to
inclusive." See fzval 8876 for its value and additional comments.
(Contributed by NM, 6-Sep-2005.)
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Theorem | fzval 8876* |
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 8877 |
An alternative way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3-Nov-2013.)
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Theorem | fzf 8878 |
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 8879 |
Membership in a finite set of sequential integers. (Contributed by NM,
21-Jul-2005.)
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Theorem | elfz 8880 |
Membership in a finite set of sequential integers. (Contributed by NM,
29-Sep-2005.)
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Theorem | elfz2 8881 |
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 8882 |
Membership in a finite set of sequential integers. (Contributed by NM,
26-Dec-2005.)
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Theorem | elfz4 8883 |
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 8884 |
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 8885 |
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 8886 |
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 8887 |
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 8888 |
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 8889 |
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 8890 |
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 8891 |
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 8892 |
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 8893 |
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 8894 |
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 8895 |
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 8896 |
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 8897 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 14-Sep-2005.)
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Theorem | elfz3 8898 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21-Jul-2005.)
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Theorem | elfz1eq 8899 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19-Sep-2005.)
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Theorem | elfzubelfz 8900 |
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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