P. 97 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eluz2 9601	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 9602	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 9603	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 9604	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 9605	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 9606	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 9607	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9608	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 9609	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 9610	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 9611	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9612	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 9613	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 9614	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9615	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 9616	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 9617	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 9618	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 9619	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 9620	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 9621	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 9622	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 9623	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 9624	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 9625	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 9626	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 9627	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 9628	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 9629	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 9630	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9631	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9632	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 9633	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 9634	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 9635	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 9636	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 9637	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 9638	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 9639	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 9640	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 9641	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 9642	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 9643	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 9644	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 9645	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 9646*	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 9647*	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 9648*	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 9649*	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 9650*	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 9651	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 9652	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 9653	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 9654	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 9655	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 9656*	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 9657*	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 9656 or uzind4ALT 9657 may be used; see comment for nnind 9000. (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 9658*	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 9659*	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 9658 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 9660*	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 9656 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 9600). (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 9661*	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 9662*	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 9663*	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 9681. (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 9664*	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 9663. (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 9665*	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 9666*	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 9667	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 9668	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 9669	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 9670	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 9671	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 9672	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 9673	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 9674	1 is not in ( ZZ>= ` 2 ). (Contributed by Paul Chapman, 21-Nov-2012.)
|- -. 1 e. ( ZZ>= ` 2 )
Theorem	elnn1uz2 9675	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 9676	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 9677*	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 )
Theorem	eluzdc 9678	Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID N e. ( ZZ>= ` M ) )
Theorem	elnn0dc 9679	Membership of an integer in NN0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN0 )
Theorem	elnndc 9680	Membership of an integer in NN is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN )
Theorem	ublbneg 9681*	The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9663. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( E. x e. RR A. y e. A y <_ x -> E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y )
Theorem	eqreznegel 9682*	Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )
Theorem	negm 9683*	The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
|- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )
Theorem	lbzbi 9684*	If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ RR -> ( E. x e. RR A. y e. A x <_ y <-> E. x e. ZZ A. y e. A x <_ y ) )
Theorem	nn01to3 9685	A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
|- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
Theorem	nn0ge2m1nnALT 9686	Alternate proof of nn0ge2m1nn 9303: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9601, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 9303. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )
4.4.12  Rational numbers (as a subset of complex numbers)
Syntax	cq 9687	Extend class notation to include the class of rationals.
class QQ
Definition	df-q 9688	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9690 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
|- QQ = ( / " ( ZZ X. NN ) )
Theorem	divfnzn 9689	Division restricted to ZZ X. NN is a function. Given excluded middle, it would be easy to prove this for CC X. ( CC \ { 0 } ). The key difference is that an element of NN is apart from zero, whereas being an element of CC \ { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN )
Theorem	elq 9690*	Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
|- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
Theorem	qmulz 9691*	If A is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
|- ( A e. QQ -> E. x e. NN ( A x. x ) e. ZZ )
Theorem	znq 9692	The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
Theorem	qre 9693	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
|- ( A e. QQ -> A e. RR )
Theorem	zq 9694	An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
|- ( A e. ZZ -> A e. QQ )
Theorem	zssq 9695	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
|- ZZ C_ QQ
Theorem	nn0ssq 9696	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN0 C_ QQ
Theorem	nnssq 9697	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN C_ QQ
Theorem	qssre 9698	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
|- QQ C_ RR
Theorem	qsscn 9699	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- QQ C_ CC
Theorem	qex 9700	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- QQ e. _V