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**Statement List for Metamath Proof Explorer -** 6701-6800 - Page 68 of 175
Type	Label	Description
Statement

Theorem	leid 6701	'Less than or equal to' is reflexive.


Theorem	ltnsym 6702	'Less than' is not symmetric.
$/\$

Theorem	ltnsym2 6703	'Less than' is antisymmetric and irreflexive. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$

Theorem	ltnsym2OLD 6704	'Less than' is antisymmetric and irreflexive.
$/\$ $/\$

Theorem	pm2.61ltlei 6705	Ordering elimination by cases.
$/\$ $/\$

Ordering on the extended reals

Theorem	elxr 6706	Membership in the set of extended reals.
$\/$ $\/$

Theorem	pnfnemnf 6707	Plus and minus infinity are distinguished elements of .


Theorem	renepnf 6708	No (finite) real equals plus infinity. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	renepnfOLD 6709	No (finite) real equals plus infinity.


Theorem	renemnf 6710	No real equals minus infinity. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	renemnfOLD 6711	No real equals minus infinity.


Theorem	renfdisj 6712	The reals and the infinities are disjoint. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	renfdisjOLD 6713	The reals and the infinities are disjoint.


Theorem	ssxr 6714	The three (non-exclusive) possibilities implied by a subset of extended reals. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$\/$ $\/$

Theorem	ssxrOLD 6715	The three (non-exclusive) possibilities implied by a subset of extended reals.
$\/$ $\/$

Theorem	xrltnr 6716	The extended real 'less than' is irreflexive.


Theorem	ltpnf 6717	Any (finite) real is less than plus infinity.


Theorem	mnflt 6718	Minus infinity is less than any (finite) real.


Theorem	mnfltpnf 6719	Minus infinity is less than plus infinity.


Theorem	mnfltxr 6720	Minus infinity is less than an extended real that is either real or plus infinity.
$\/$

Theorem	pnfnlt 6721	No extended real is greater than plus infinity.


Theorem	nltmnf 6722	No extended real is less than minus infinity.


Theorem	pnfge 6723	Plus infinity is an upper bound for extended reals.


Theorem	mnfle 6724	Minus infinity is less than or equal to any extended real.


Theorem	xrltnsym 6725	Ordering on the extended reals is not symmetric.
$/\$

Theorem	xrltnsym2 6726	'Less than' is antisymmetric and irreflexive for extended reals.
$/\$ $/\$

Theorem	xrlttri 6727	Ordering on the extended reals satisfies strict trichotomy.
$/\$ $\/$

Theorem	xrlttr 6728	Ordering on the extended reals is transitive.
$/\$ $/\$ $/\$

Theorem	xrltso 6729	'Less than' is a strict ordering on the extended reals.


Theorem	xrlttri2 6730	Trichotomy law for 'less than' for extended reals.
$/\$ $\/$

Theorem	xrlttri3 6731	Trichotomy law for 'less than' for extended reals.
$/\$ $/\$

Theorem	xrleloe 6732	'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals.
$/\$ $\/$

Theorem	xrleltne 6733	'Less than or equal to' implies 'less than' is not 'equals', for extended reals.
$/\$ $/\$

Theorem	xrltle 6734	'Less than' implies 'less than or equal' for extended reals.
$/\$

Theorem	xrleid 6735	'Less than or equal to' is reflexive for extended reals.


Theorem	xrletri 6736	Trichotomy law for extended reals.
$/\$ $\/$

Theorem	xrlelttr 6737	Transitive law for ordering on extended reals.
$/\$ $/\$ $/\$

Theorem	xrltletr 6738	Transitive law for ordering on extended reals.
$/\$ $/\$ $/\$

Theorem	xrletr 6739	Transitive law for ordering on extended reals.
$/\$ $/\$ $/\$

Theorem	xrltne 6740	'Less than' implies not equal for extended reals.
$/\$ $/\$

Theorem	nltpnft 6741	An extended real is not less than plus infinity iff they are equal.


Theorem	ngtmnft 6742	An extended real is not greater than minus infinity iff they are equal.


Theorem	xrrebnd 6743	An extended real is real iff it is strictly bounded by infinities.
$/\$

Theorem	xrre 6744	A way of proving that an extended real is real.
$/\$ $/\$ $/\$

Theorem	xrre2 6745	An extended real between two others is real.
$/\$ $/\$ $/\$ $/\$

Ordering on reals (cont.)

Theorem	eqle 6746	Equality implies 'less than or equal to'.
$/\$

Theorem	lttri2i 6747	Consequence of trichotomy.
$\/$

Theorem	lttri3i 6748	Consequence of trichotomy.
$/\$

Theorem	letri3i 6749	Consequence of trichotomy.
$/\$

Theorem	leloei 6750	'Less than or equal to' in terms of 'less than'.
$\/$

Theorem	ltleni 6751	'Less than' expressed in terms of 'less than or equal to'.
$/\$

Theorem	ltnsymi 6752	'Less than' is not symmetric.


Theorem	lenlti 6753	'Less than or equal to' in terms of 'less than'.


Theorem	ltnlei 6754	'Less than' in terms of 'less than or equal to'.


Theorem	ltlei 6755	'Less than' implies 'less than or equal to'.


Theorem	ltleii 6756	'Less than' implies 'less than or equal to' (inference).


Theorem	eqlei 6757	Equality implies 'less than or equal to'.


Theorem	ltnei 6758	'Less than' implies not equal.


Theorem	letrii 6759	Trichotomy law for 'less than or equal to'.
$\/$

Theorem	lttri 6760	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20.
$/\$

Theorem	lelttri 6761	'Less than or equal to', 'less than' transitive law.
$/\$

Theorem	ltletri 6762	'Less than', 'less than or equal to' transitive law.
$/\$

Theorem	letri 6763	'Less than or equal to' is transitive.
$/\$

Theorem	le2tri3i 6764	Extended trichotomy law for 'less than or equal to'.
$/\$ $/\$ $/\$ $/\$

Theorem	ltadd2i 6765	Addition to both sides of 'less than'. (Proof shortened by Paul Chapman, 27-Jan-2008.)


Theorem	ltadd1i 6766	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.


Theorem	leadd1i 6767	Addition to both sides of 'less than or equal to'.


Theorem	leadd2i 6768	Addition to both sides of 'less than or equal to'.


Theorem	ltsubaddi 6769	'Less than' relationship between subtraction and addition. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	ltsubaddiOLD 6770	'Less than' relationship between subtraction and addition.


Theorem	lesubaddi 6771	'Less than or equal to' relationship between subtraction and addition. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	lesubaddiOLD 6772	'Less than or equal to' relationship between subtraction and addition.


Theorem	lt2addi 6773	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20.
$/\$

Theorem	le2addi 6774	Adding both side of two inequalities.
$/\$

Theorem	addgt0i 6775	Addition of 2 positive numbers is positive. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addgt0iOLD 6776	Addition of 2 positive numbers is positive.
$/\$

Theorem	addge0i 6777	Addition of 2 nonnegative numbers is nonnegative. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addge0iOLD 6778	Addition of 2 nonnegative numbers is nonnegative.
$/\$

Theorem	addgegt0i 6779	Addition of nonnegative and positive numbers is positive. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addgegt0iOLD 6780	Addition of nonnegative and positive numbers is positive.
$/\$

Theorem	addgt0ii 6781	Addition of 2 positive numbers is positive.


Theorem	add20i 6782	Two nonnegative numbers are zero iff their sum is zero.
$/\$ $/\$

Theorem	ltnegi 6783	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.


Theorem	lenegi 6784	Negative of both sides of 'less than or equal to'.


Theorem	ltnegcon2i 6785	Contraposition of negative in 'less than'.


Theorem	mulgt0i 6786	The product of two positive numbers is positive.
$/\$

Theorem	mulge0i 6787	The product of two nonnegative numbers is nonnegative.
$/\$

Theorem	mulgt0ii 6788	The product of two positive numbers is positive.


Theorem	ltnri 6789	'Less than' is irreflexive.


Theorem	leidi 6790	'Less than or equal to' is reflexive.


Theorem	gt0ne0i 6791	Positive means nonzero (useful for ordering theorems involving division).


Theorem	lesub0i 6792	Lemma to show a nonnegative number is zero. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	lesub0iOLD 6793	Lemma to show a nonnegative number is zero.
$/\$

Theorem	msqgt0i 6794	A nonzero square is positive. Theorem I.20 of [Apostol] p. 20.


Theorem	msqge0i 6795	A square is nonnegative. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	msqge0iOLD 6796	A square is nonnegative.


Theorem	msqgt0 6797	A nonzero square is positive. Theorem I.20 of [Apostol] p. 20.
$/\$

Theorem	msqge0 6798	A square is nonnegative.


Theorem	gt0ne0ii 6799	Positive implies nonzero.


Theorem	gt0ne0 6800	Positive implies nonzero.
$/\$

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