P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zex 9401	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 9402	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 9403	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 9404	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 9405	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 9406	Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
|- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
Theorem	elznn0 9407	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
Theorem	elznn 9408	Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )
Theorem	nnssz 9409	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 9410	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 9411	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 9412	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 9413	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 9414	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ
Theorem	elnnz1 9415	Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( N e. NN <-> ( N e. ZZ /\ 1 <_ N ) )
Theorem	nnzrab 9416	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN = { x e. ZZ | 1 <_ x }
Theorem	nn0zrab 9417	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN0 = { x e. ZZ | 0 <_ x }
Theorem	1z 9418	One is an integer. (Contributed by NM, 10-May-2004.)
|- 1 e. ZZ
Theorem	1zzd 9419	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. ZZ )
Theorem	2z 9420	Two is an integer. (Contributed by NM, 10-May-2004.)
|- 2 e. ZZ
Theorem	3z 9421	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. ZZ
Theorem	4z 9422	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|- 4 e. ZZ
Theorem	znegcl 9423	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> -u N e. ZZ )
Theorem	neg1z 9424	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 9425	A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. CC -> ( A e. ZZ <-> -u A e. ZZ ) )
Theorem	nn0negz 9426	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 9427	The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- -u N e. ZZ
Theorem	peano2z 9428	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 9429	Lemma for zaddcl 9432. Special case in which N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( M + N ) e. ZZ )
Theorem	peano2zm 9430	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 9431	Lemma for zaddcl 9432. Special case in which -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )
Theorem	zaddcl 9432	Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
Theorem	zsubcl 9433	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 9434	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 9435	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 9436	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 9437	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 9438	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> -. B <_ A ) )
Theorem	zleloe 9439	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
Theorem	znnnlt1 9440	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
|- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) )
Theorem	nnnle0 9441	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
|- ( A e. NN -> -. A <_ 0 )
Theorem	zletr 9442	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
|- ( ( J e. ZZ /\ K e. ZZ /\ L e. ZZ ) -> ( ( J <_ K /\ K <_ L ) -> J <_ L ) )
Theorem	zrevaddcl 9443	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
|- ( N e. ZZ -> ( ( M e. CC /\ ( M + N ) e. ZZ ) <-> M e. ZZ ) )
Theorem	znnsub 9444	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9095.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 9445	The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( A + B ) e. ( RR \ ZZ ) )
Theorem	zmulcl 9446	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Theorem	zltp1le 9447	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	zleltp1 9448	Integer ordering relation. (Contributed by NM, 10-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	zlem1lt 9449	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	zltlem1 9450	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	zgt0ge1 9451	An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
|- ( Z e. ZZ -> ( 0 < Z <-> 1 <_ Z ) )
Theorem	nnleltp1 9452	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( A <_ B <-> A < ( B + 1 ) ) )
Theorem	nnltp1le 9453	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( A + 1 ) <_ B ) )
Theorem	nnaddm1cl 9454	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( ( A + B ) - 1 ) e. NN )
Theorem	nn0ltp1le 9455	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	nn0leltp1 9456	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	nn0ltlem1 9457	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	znn0sub 9458	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9459.) (Contributed by NM, 14-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	nn0sub 9459	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	ltsubnn0 9460	Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> ( A - B ) e. NN0 ) )
Theorem	nn0negleid 9461	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
|- ( A e. NN0 -> -u A <_ A )
Theorem	difgtsumgt 9462	If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
|- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) -> C < ( A + B ) ) )
Theorem	nn0n0n1ge2 9463	A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
Theorem	elz2 9464*	Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)
|- ( N e. ZZ <-> E. x e. NN E. y e. NN N = ( x - y ) )
Theorem	dfz2 9465	Alternate definition of the integers, based on elz2 9464. (Contributed by Mario Carneiro, 16-May-2014.)
|- ZZ = ( - " ( NN X. NN ) )
Theorem	nn0sub2 9466	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 )
Theorem	zapne 9467	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M # N <-> M =/= N ) )
Theorem	zdceq 9468	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
Theorem	zdcle 9469	Integer <_ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B )
Theorem	zdclt 9470	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
Theorem	zltlen 9471	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8725 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) )
Theorem	nn0n0n1ge2b 9472	A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
|- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )
Theorem	nn0lt10b 9473	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN0 -> ( N < 1 <-> N = 0 ) )
Theorem	nn0lt2 9474	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
|- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) )
Theorem	nn0le2is012 9475	A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
|- ( ( N e. NN0 /\ N <_ 2 ) -> ( N = 0 \/ N = 1 \/ N = 2 ) )
Theorem	nn0lem1lt 9476	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnlem1lt 9477	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnltlem1 9478	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	nnm1ge0 9479	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
|- ( N e. NN -> 0 <_ ( N - 1 ) )
Theorem	nn0ge0div 9480	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ L e. NN ) -> 0 <_ ( K / L ) )
Theorem	zdiv 9481*	Two ways to express " M divides N. (Contributed by NM, 3-Oct-2008.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )
Theorem	zdivadd 9482	Property of divisibility: if D divides A and B then it divides A + B. (Contributed by NM, 3-Oct-2008.)
|- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ )
Theorem	zdivmul 9483	Property of divisibility: if D divides A then it divides B x. A. (Contributed by NM, 3-Oct-2008.)
|- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( A / D ) e. ZZ ) -> ( ( B x. A ) / D ) e. ZZ )
Theorem	zextle 9484*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k <_ M <-> k <_ N ) ) -> M = N )
Theorem	zextlt 9485*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k < M <-> k < N ) ) -> M = N )
Theorem	recnz 9486	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. RR /\ 1 < A ) -> -. ( 1 / A ) e. ZZ )
Theorem	btwnnz 9487	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. ZZ /\ A < B /\ B < ( A + 1 ) ) -> -. B e. ZZ )
Theorem	gtndiv 9488	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ )
Theorem	halfnz 9489	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|- -. ( 1 / 2 ) e. ZZ
Theorem	3halfnz 9490	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|- -. ( 3 / 2 ) e. ZZ
Theorem	suprzclex 9491*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
|- ( 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_ ZZ )   =>    |- ( ph -> sup ( A , RR , < ) e. A )
Theorem	prime 9492*	Two ways to express " A is a prime number (or 1)". (Contributed by NM, 4-May-2005.)
|- ( A e. NN -> ( A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )
Theorem	msqznn 9493	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
|- ( ( A e. ZZ /\ A =/= 0 ) -> ( A x. A ) e. NN )
Theorem	zneo 9494	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )
Theorem	nneoor 9495	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
|- ( N e. NN -> ( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	nneo 9496	A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
|- ( N e. NN -> ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	nneoi 9497	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
|- N e. NN   =>    |- ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN )
Theorem	zeo 9498	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zeo2 9499	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
|- ( N e. ZZ -> ( ( N / 2 ) e. ZZ <-> -. ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	peano2uz2 9500*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
|- ( ( A e. ZZ /\ B e. { x e. ZZ | A <_ x } ) -> ( B + 1 ) e. { x e. ZZ | A <_ x } )