Theorem List for Intuitionistic Logic Explorer - 9601-9700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | elrpd 9601 |
Membership in the set of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | nnrpd 9602 |
A positive integer is a positive real. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zgt1rpn0n1 9603 |
An integer greater than 1 is a positive real number not equal to 0 or 1.
Useful for working with integer logarithm bases (which is a common case,
e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux,
26-Sep-2017.) (Proof shortened by AV, 9-Jul-2022.)
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Theorem | rpred 9604 |
A positive real is a real. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | rpxrd 9605 |
A positive real is an extended real. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | rpcnd 9606 |
A positive real is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | rpgt0d 9607 |
A positive real is greater than zero. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | rpge0d 9608 |
A positive real is greater than or equal to zero. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rpne0d 9609 |
A positive real is nonzero. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | rpap0d 9610 |
A positive real is apart from zero. (Contributed by Jim Kingdon,
28-Jul-2021.)
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Theorem | rpregt0d 9611 |
A positive real is real and greater than zero. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rprege0d 9612 |
A positive real is real and greater or equal to zero. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rprene0d 9613 |
A positive real is a nonzero real number. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rpcnne0d 9614 |
A positive real is a nonzero complex number. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rpreccld 9615 |
Closure law for reciprocation of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rprecred 9616 |
Closure law for reciprocation of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rphalfcld 9617 |
Closure law for half of a positive real. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | reclt1d 9618 |
The reciprocal of a positive number less than 1 is greater than 1.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | recgt1d 9619 |
The reciprocal of a positive number greater than 1 is less than 1.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | rpaddcld 9620 |
Closure law for addition of positive reals. Part of Axiom 7 of
[Apostol] p. 20. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rpmulcld 9621 |
Closure law for multiplication of positive reals. Part of Axiom 7 of
[Apostol] p. 20. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | rpdivcld 9622 |
Closure law for division of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltrecd 9623 |
The reciprocal of both sides of 'less than'. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | lerecd 9624 |
The reciprocal of both sides of 'less than or equal to'. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltrec1d 9625 |
Reciprocal swap in a 'less than' relation. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | lerec2d 9626 |
Reciprocal swap in a 'less than or equal to' relation. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | lediv2ad 9627 |
Division of both sides of 'less than or equal to' into a nonnegative
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdiv2d 9628 |
Division of a positive number by both sides of 'less than'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lediv2d 9629 |
Division of a positive number by both sides of 'less than or equal to'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivdivd 9630 |
Invert ratios of positive numbers and swap their ordering.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | divge1 9631 |
The ratio of a number over a smaller positive number is larger than 1.
(Contributed by Glauco Siliprandi, 5-Apr-2020.)
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Theorem | divlt1lt 9632 |
A real number divided by a positive real number is less than 1 iff the
real number is less than the positive real number. (Contributed by AV,
25-May-2020.)
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Theorem | divle1le 9633 |
A real number divided by a positive real number is less than or equal to 1
iff the real number is less than or equal to the positive real number.
(Contributed by AV, 29-Jun-2021.)
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Theorem | ledivge1le 9634 |
If a number is less than or equal to another number, the number divided by
a positive number greater than or equal to one is less than or equal to
the other number. (Contributed by AV, 29-Jun-2021.)
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Theorem | ge0p1rpd 9635 |
A nonnegative number plus one is a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rerpdivcld 9636 |
Closure law for division of a real by a positive real. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | ltsubrpd 9637 |
Subtracting a positive real from another number decreases it.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltaddrpd 9638 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltaddrp2d 9639 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | ltmulgt11d 9640 |
Multiplication by a number greater than 1. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltmulgt12d 9641 |
Multiplication by a number greater than 1. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | gt0divd 9642 |
Division of a positive number by a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | ge0divd 9643 |
Division of a nonnegative number by a positive number. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rpgecld 9644 |
A number greater or equal to a positive real is positive real.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | divge0d 9645 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul1d 9646 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul2d 9647 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | lemul1d 9648 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemul2d 9649 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdiv1d 9650 |
Division of both sides of 'less than' by a positive number.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lediv1d 9651 |
Division of both sides of a less than or equal to relation by a positive
number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmuldivd 9652 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmuldiv2d 9653 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemuldivd 9654 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lemuldiv2d 9655 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | ltdivmuld 9656 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltdivmul2d 9657 |
'Less than' relationship between division and multiplication.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivmuld 9658 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ledivmul2d 9659 |
'Less than or equal to' relationship between division and
multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul1dd 9660 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | ltmul2dd 9661 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by Mario Carneiro,
30-May-2016.)
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Theorem | ltdiv1dd 9662 |
Division of both sides of 'less than' by a positive number.
(Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lediv1dd 9663 |
Division of both sides of a less than or equal to relation by a
positive number. (Contributed by Mario Carneiro, 30-May-2016.)
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Theorem | lediv12ad 9664 |
Comparison of ratio of two nonnegative numbers. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltdiv23d 9665 |
Swap denominator with other side of 'less than'. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | lediv23d 9666 |
Swap denominator with other side of 'less than or equal to'.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | mul2lt0rlt0 9667 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0rgt0 9668 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0llt0 9669 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
19-Sep-2018.)
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Theorem | mul2lt0lgt0 9670 |
If the result of a multiplication is strictly negative, then
multiplicands are of different signs. (Contributed by Thierry Arnoux,
2-Oct-2018.)
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Theorem | mul2lt0np 9671 |
The product of multiplicands of different signs is negative.
(Contributed by Jim Kingdon, 25-Feb-2024.)
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Theorem | mul2lt0pn 9672 |
The product of multiplicands of different signs is negative.
(Contributed by Jim Kingdon, 25-Feb-2024.)
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Theorem | lt2mul2divd 9673 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | nnledivrp 9674 |
Division of a positive integer by a positive number is less than or equal
to the integer iff the number is greater than or equal to 1. (Contributed
by AV, 19-Jun-2021.)
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Theorem | nn0ledivnn 9675 |
Division of a nonnegative integer by a positive integer is less than or
equal to the integer. (Contributed by AV, 19-Jun-2021.)
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Theorem | addlelt 9676 |
If the sum of a real number and a positive real number is less than or
equal to a third real number, the first real number is less than the third
real number. (Contributed by AV, 1-Jul-2021.)
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4.5.2 Infinity and the extended real number
system (cont.)
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Syntax | cxne 9677 |
Extend class notation to include the negative of an extended real.
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Syntax | cxad 9678 |
Extend class notation to include addition of extended reals.
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Syntax | cxmu 9679 |
Extend class notation to include multiplication of extended reals.
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Definition | df-xneg 9680 |
Define the negative of an extended real number. (Contributed by FL,
26-Dec-2011.)
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Definition | df-xadd 9681* |
Define addition over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Definition | df-xmul 9682* |
Define multiplication over extended real numbers. (Contributed by Mario
Carneiro, 20-Aug-2015.)
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Theorem | ltxr 9683 |
The 'less than' binary relation on the set of extended reals.
Definition 12-3.1 of [Gleason] p. 173.
(Contributed by NM,
14-Oct-2005.)
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Theorem | elxr 9684 |
Membership in the set of extended reals. (Contributed by NM,
14-Oct-2005.)
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Theorem | xrnemnf 9685 |
An extended real other than minus infinity is real or positive infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrnepnf 9686 |
An extended real other than plus infinity is real or negative infinite.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | xrltnr 9687 |
The extended real 'less than' is irreflexive. (Contributed by NM,
14-Oct-2005.)
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Theorem | ltpnf 9688 |
Any (finite) real is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | ltpnfd 9689 |
Any (finite) real is less than plus infinity. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
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Theorem | 0ltpnf 9690 |
Zero is less than plus infinity (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnflt 9691 |
Minus infinity is less than any (finite) real. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnflt0 9692 |
Minus infinity is less than 0 (common case). (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | mnfltpnf 9693 |
Minus infinity is less than plus infinity. (Contributed by NM,
14-Oct-2005.)
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Theorem | mnfltxr 9694 |
Minus infinity is less than an extended real that is either real or plus
infinity. (Contributed by NM, 2-Feb-2006.)
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Theorem | pnfnlt 9695 |
No extended real is greater than plus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | nltmnf 9696 |
No extended real is less than minus infinity. (Contributed by NM,
15-Oct-2005.)
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Theorem | pnfge 9697 |
Plus infinity is an upper bound for extended reals. (Contributed by NM,
30-Jan-2006.)
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Theorem | 0lepnf 9698 |
0 less than or equal to positive infinity. (Contributed by David A.
Wheeler, 8-Dec-2018.)
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Theorem | nn0pnfge0 9699 |
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.)
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Theorem | mnfle 9700 |
Minus infinity is less than or equal to any extended real. (Contributed
by NM, 19-Jan-2006.)
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