P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	9t8e72 9701	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 8 ) = ; 7 2
Theorem	9t9e81 9702	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 9 ) = ; 8 1
Theorem	9t11e99 9703	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
|- ( 9 x. ; 1 1 ) = ; 9 9
Theorem	9lt10 9704	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 9 < ; 1 0
Theorem	8lt10 9705	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 8 < ; 1 0
Theorem	7lt10 9706	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 7 < ; 1 0
Theorem	6lt10 9707	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 6 < ; 1 0
Theorem	5lt10 9708	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 5 < ; 1 0
Theorem	4lt10 9709	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 4 < ; 1 0
Theorem	3lt10 9710	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 3 < ; 1 0
Theorem	2lt10 9711	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 2 < ; 1 0
Theorem	1lt10 9712	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 1 < ; 1 0
Theorem	decbin0 9713	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) )
Theorem	decbin2 9714	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )
Theorem	decbin3 9715	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( ( 4 x. A ) + 3 ) = ( ( 2 x. ( ( 2 x. A ) + 1 ) ) + 1 )
Theorem	halfthird 9716	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
Theorem	5recm6rec 9717	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
|- ( ( 1 / 5 ) - ( 1 / 6 ) ) = ( 1 / ; 3 0 )
4.4.11  Upper sets of integers
Syntax	cuz 9718	Extend class notation with the upper integer function. Read " ZZ>= ` M " as "the set of integers greater than or equal to M".
class ZZ>=
Definition	df-uz 9719*	Define a function whose value at j is the semi-infinite set of contiguous integers starting at j, which we will also call the upper integers starting at j. Read " ZZ>= ` M " as "the set of integers greater than or equal to M". See uzval 9720 for its value, uzssz 9738 for its relationship to ZZ, nnuz 9754 and nn0uz 9753 for its relationships to NN and NN0, and eluz1 9722 and eluz2 9724 for its membership relations. (Contributed by NM, 5-Sep-2005.)
|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
Theorem	uzval 9720*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ZZ -> ( ZZ>= ` N ) = { k e. ZZ | N <_ k } )
Theorem	uzf 9721	The domain and codomain of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ZZ>= : ZZ --> ~P ZZ
Theorem	eluz1 9722	Membership in the upper set of integers starting at M. (Contributed by NM, 5-Sep-2005.)
|- ( M e. ZZ -> ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) )
Theorem	eluzel2 9723	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
Theorem	eluz2 9724	Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show M e. ZZ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
Theorem	eluz1i 9725	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
|- M e. ZZ   =>    |- ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) )
Theorem	eluzuzle 9726	An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
|- ( ( B e. ZZ /\ B <_ A ) -> ( C e. ( ZZ>= ` A ) -> C e. ( ZZ>= ` B ) ) )
Theorem	eluzelz 9727	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
Theorem	eluzelre 9728	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
|- ( N e. ( ZZ>= ` M ) -> N e. RR )
Theorem	eluzelcn 9729	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( N e. ( ZZ>= ` M ) -> N e. CC )
Theorem	eluzle 9730	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9731	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
Theorem	uzid 9732	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
Theorem	uzidd 9733	Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> M e. ZZ )   =>    |- ( ph -> M e. ( ZZ>= ` M ) )
Theorem	uzn0 9734	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9735	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
|- ( ( M e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` N ) ) -> M e. ( ZZ>= ` N ) )
Theorem	uztrn2 9736	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` K )   =>    |- ( ( N e. Z /\ M e. ( ZZ>= ` N ) ) -> M e. Z )
Theorem	uzneg 9737	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9738	An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( ZZ>= ` M ) C_ ZZ
Theorem	uzss 9739	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
Theorem	uztric 9740	Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
Theorem	uz11 9741	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
|- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )
Theorem	eluzp1m1 9742	Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
Theorem	eluzp1l 9743	Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N )
Theorem	eluzp1p1 9744	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) )
Theorem	eluzaddi 9745	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &    |- K e. ZZ   =>    |- ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )
Theorem	eluzsubi 9746	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &    |- K e. ZZ   =>    |- ( N e. ( ZZ>= ` ( M + K ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Theorem	eluzadd 9747	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )
Theorem	eluzsub 9748	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Theorem	uzm1 9749	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
Theorem	uznn0sub 9750	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 )
Theorem	uzin 9751	Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ZZ>= ` M ) i^i ( ZZ>= ` N ) ) = ( ZZ>= ` if ( M <_ N , N , M ) ) )
Theorem	uzp1 9752	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
Theorem	nn0uz 9753	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9754	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9755	A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
|- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
Theorem	elnn0uz 9756	A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
|- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
Theorem	eluz2nn 9757	An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
|- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
Theorem	eluz4eluz2 9758	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) )
Theorem	eluz4nn 9759	An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. NN )
Theorem	eluzge2nn0 9760	If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
|- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
Theorem	eluz2n0 9761	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> N =/= 0 )
Theorem	uzuzle23 9762	An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( A e. ( ZZ>= ` 3 ) -> A e. ( ZZ>= ` 2 ) )
Theorem	eluzge3nn 9763	If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
Theorem	uz3m2nn 9764	An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
Theorem	1eluzge0 9765	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- 1 e. ( ZZ>= ` 0 )
Theorem	2eluzge0 9766	2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- 2 e. ( ZZ>= ` 0 )
Theorem	2eluzge1 9767	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- 2 e. ( ZZ>= ` 1 )
Theorem	uznnssnn 9768	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
|- ( N e. NN -> ( ZZ>= ` N ) C_ NN )
Theorem	raluz 9769*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( M e. ZZ -> ( A. n e. ( ZZ>= ` M ) ph <-> A. n e. ZZ ( M <_ n -> ph ) ) )
Theorem	raluz2 9770*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( A. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ -> A. n e. ZZ ( M <_ n -> ph ) ) )
Theorem	rexuz 9771*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ph <-> E. n e. ZZ ( M <_ n /\ ph ) ) )
Theorem	rexuz2 9772*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( E. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) )
Theorem	2rexuz 9773*	Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
|- ( E. m E. n e. ( ZZ>= ` m ) ph <-> E. m e. ZZ E. n e. ZZ ( m <_ n /\ ph ) )
Theorem	peano2uz 9774	Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
Theorem	peano2uzs 9775	Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( N + 1 ) e. Z )
Theorem	peano2uzr 9776	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
Theorem	uzaddcl 9777	Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
|- ( ( N e. ( ZZ>= ` M ) /\ K e. NN0 ) -> ( N + K ) e. ( ZZ>= ` M ) )
Theorem	nn0pzuz 9778	The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
|- ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) )
Theorem	uzind4 9779*	Induction on the upper set of integers that starts at an integer M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ( M e. ZZ -> ps )   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	uzind4ALT 9780*	Induction on the upper set of integers that starts at an integer M. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9779 or uzind4ALT 9780 may be used; see comment for nnind 9122. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( M e. ZZ -> ps )   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   &    |- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	uzind4s 9781*	Induction on the upper set of integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
|- ( M e. ZZ -> [. M / k ]. ph )   &    |- ( k e. ( ZZ>= ` M ) -> ( ph -> [. ( k + 1 ) / k ]. ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )
Theorem	uzind4s2 9782*	Induction on the upper set of integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 9781 when j and k must be distinct in [. ( k + 1 ) / j ]. ph. (Contributed by NM, 16-Nov-2005.)
|- ( M e. ZZ -> [. M / j ]. ph )   &    |- ( k e. ( ZZ>= ` M ) -> ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph )
Theorem	uzind4i 9783*	Induction on the upper integers that start at M. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9779 assuming that ps holds unconditionally. Notice that N e. ( ZZ>= ` M ) implies that the lower bound M is an integer ( M e. ZZ, see eluzel2 9723). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ps   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	indstr 9784*	Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. NN -> ( A. y e. NN ( y < x -> ps ) -> ph ) )   =>    |- ( x e. NN -> ph )
Theorem	infrenegsupex 9785*	The infimum of a set of reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
Theorem	supinfneg 9786*	If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9804. (Contributed by Jim Kingdon, 15-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. A } -. y < x /\ A. y e. RR ( x < y -> E. z e. { w e. RR | -u w e. A } z < y ) ) )
Theorem	infsupneg 9787*	If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9786. (Contributed by Jim Kingdon, 15-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. A } -. x < y /\ A. y e. RR ( y < x -> E. z e. { w e. RR | -u w e. A } y < z ) ) )
Theorem	supminfex 9788*	A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )
Theorem	infregelbex 9789*	Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( B <_ inf ( A , RR , < ) <-> A. z e. A B <_ z ) )
Theorem	eluznn0 9790	Membership in a nonnegative upper set of integers implies membership in NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
Theorem	eluznn 9791	Membership in a positive upper set of integers implies membership in NN. (Contributed by JJ, 1-Oct-2018.)
|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
Theorem	eluz2b1 9792	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) )
Theorem	eluz2gt1 9793	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
Theorem	eluz2b2 9794	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ 1 < N ) )
Theorem	eluz2b3 9795	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
Theorem	uz2m1nn 9796	One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
Theorem	1nuz2 9797	1 is not in ( ZZ>= ` 2 ). (Contributed by Paul Chapman, 21-Nov-2012.)
|- -. 1 e. ( ZZ>= ` 2 )
Theorem	elnn1uz2 9798	A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
Theorem	uz2mulcl 9799	Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> ( M x. N ) e. ( ZZ>= ` 2 ) )
Theorem	indstr2 9800*	Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
|- ( x = 1 -> ( ph <-> ch ) )   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ch   &    |- ( x e. ( ZZ>= ` 2 ) -> ( A. y e. NN ( y < x -> ps ) -> ph ) )   =>    |- ( x e. NN -> ph )