Theorem List for Intuitionistic Logic Explorer - 9701-9800 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | ioopos 9701 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
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Theorem | ioorp 9702 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
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Theorem | iooshf 9703 |
Shift the arguments of the open interval function. (Contributed by NM,
17-Aug-2008.)
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Theorem | iocssre 9704 |
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 9705 |
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 9706 |
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 9707 |
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 9708 |
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 9709 |
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 9710 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
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Theorem | icossicc 9711 |
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 9712 |
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 9713 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
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Theorem | iocssioo 9714 |
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 9715 |
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 9716 |
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 9717* |
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 9718 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elioomnf 9719 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
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Theorem | elicopnf 9720 |
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16-Sep-2014.)
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Theorem | repos 9721 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
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Theorem | ioof 9722 |
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 9723 |
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 9724 |
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 9725* |
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 9726 |
Open intervals are elements of the set of all open intervals.
(Contributed by Jim Kingdon, 4-Apr-2020.)
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Theorem | elrege0 9727 |
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 9728 |
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 9729 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
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Theorem | 0e0icopnf 9730 |
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | 0e0iccpnf 9731 |
0 is a member of
(common case). (Contributed by David
A. Wheeler, 8-Dec-2018.)
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Theorem | ge0addcl 9732 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
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Theorem | ge0mulcl 9733 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
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Theorem | ge0xaddcl 9734 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 26-Aug-2015.)
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Theorem | lbicc2 9735 |
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 9736 |
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 9737 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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Theorem | 1elunit 9738 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
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Theorem | iooneg 9739 |
Membership in a negated open real interval. (Contributed by Paul Chapman,
26-Nov-2007.)
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Theorem | iccneg 9740 |
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26-Nov-2007.)
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Theorem | icoshft 9741 |
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 9742* |
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 9743 |
End-to-end closed-below, open-above real intervals are disjoint.
(Contributed by Mario Carneiro, 16-Jun-2014.)
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Theorem | ioodisj 9744 |
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 9745 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftri 9746 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftl 9747 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccshftli 9748 |
Membership in a shifted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccdil 9749 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | iccdili 9750 |
Membership in a dilated interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | icccntr 9751 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | icccntri 9752 |
Membership in a contracted interval. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | divelunit 9753 |
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 9754 |
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 9755* |
Describe a bijection from to an arbitrary nontrivial
closed interval . (Contributed by Mario Carneiro,
8-Sep-2015.)
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Theorem | unitssre 9756 |
is a subset of the reals.
(Contributed by David Moews,
28-Feb-2017.)
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Theorem | zltaddlt1le 9757 |
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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4.5.4 Finite intervals of integers
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Syntax | cfz 9758 |
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 9759* |
Define an operation that produces a finite set of sequential integers.
Read " " as "the set of integers from
to
inclusive." See fzval 9760 for its value and additional comments.
(Contributed by NM, 6-Sep-2005.)
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Theorem | fzval 9760* |
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 9761 |
An alternate way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3-Nov-2013.)
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Theorem | fzf 9762 |
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 9763 |
Membership in a finite set of sequential integers. (Contributed by NM,
21-Jul-2005.)
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Theorem | elfz 9764 |
Membership in a finite set of sequential integers. (Contributed by NM,
29-Sep-2005.)
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Theorem | elfz2 9765 |
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 9766 |
Membership in a finite set of sequential integers. (Contributed by NM,
26-Dec-2005.)
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Theorem | elfz4 9767 |
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 9768 |
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 9769 |
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 9770 |
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 9771 |
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 9772 |
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 9773 |
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 9774 |
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 9775 |
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 9776 |
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 9777 |
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 9778 |
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 9779 |
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 9780 |
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 9781 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 14-Sep-2005.)
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Theorem | elfz3 9782 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21-Jul-2005.)
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Theorem | elfz1eq 9783 |
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19-Sep-2005.)
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Theorem | elfzubelfz 9784 |
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 9785 |
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 9786* |
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15-Apr-2020.)
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Theorem | fztri3or 9787 |
Trichotomy in terms of a finite interval of integers. (Contributed by Jim
Kingdon, 1-Jun-2020.)
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Theorem | fzdcel 9788 |
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 9789 |
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 9790 |
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 9791 |
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 9792 |
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 9793 |
Two ways to say a finite 1-based sequence is empty. (Contributed by Paul
Chapman, 26-Oct-2012.)
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Theorem | fz10 9794 |
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 9795 |
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 9796 |
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 9797 |
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 9798 |
Split a finite interval of integers into two parts. (Contributed by
Mario Carneiro, 13-Apr-2016.)
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Theorem | fzsplit 9799 |
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 9800 |
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17-Jun-2010.)
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