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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelz 9201 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )
 ) )
 
Theoremnnnegz 9202 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 9203 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 9204 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 9205 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 9206 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 9207 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 9208 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 9209 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 9210 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theorem0zd 9211 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  ZZ )
 
Theoremelnn0z 9212 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 9213 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 ) ) )
 
Theoremelznn0 9214 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 ) ) )
 
Theoremelznn 9215 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 )
 ) )
 
Theoremnnssz 9216 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 9217 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 9218 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 9219 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 9220 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 9221 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 9222 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 ) )
 
Theoremnnzrab 9223 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 9224 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 9225 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem1zzd 9226 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  ZZ )
 
Theorem2z 9227 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theorem3z 9228 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  ZZ
 
Theorem4z 9229 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
 |-  4  e.  ZZ
 
Theoremznegcl 9230 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremneg1z 9231 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  ZZ
 
Theoremznegclb 9232 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 ) )
 
Theoremnn0negz 9233 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  -u N  e.  ZZ )
 
Theoremnn0negzi 9234 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  -u N  e.  ZZ
 
Theorempeano2z 9235 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
 |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
 
Theoremzaddcllempos 9236 Lemma for zaddcl 9239. 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 )
 
Theorempeano2zm 9237 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( N  e.  ZZ  ->  ( N  -  1
 )  e.  ZZ )
 
Theoremzaddcllemneg 9238 Lemma for zaddcl 9239. 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 )
 
Theoremzaddcl 9239 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 )
 
Theoremzsubcl 9240 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
 
Theoremztri3or0 9241 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( N  e.  ZZ  ->  ( N  <  0  \/  N  =  0  \/  0  <  N ) )
 
Theoremztri3or 9242 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
 
Theoremzletric 9243 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremzlelttric 9244 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremzltnle 9245 '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 )
 )
 
Theoremzleloe 9246 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 ) ) )
 
Theoremznnnlt1 9247 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 ) )
 
Theoremzletr 9248 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 ) )
 
Theoremzrevaddcl 9249 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 )
 )
 
Theoremznnsub 9250 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8904.) (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( N  -  M )  e.  NN )
 )
 
Theoremnzadd 9251 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 ) )
 
Theoremzmulcl 9252 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
 
Theoremzltp1le 9253 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 )
 )
 
Theoremzleltp1 9254 Integer ordering relation. (Contributed by NM, 10-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremzlem1lt 9255 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremzltlem1 9256 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremzgt0ge1 9257 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 ) )
 
Theoremnnleltp1 9258 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 ) ) )
 
Theoremnnltp1le 9259 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( A  +  1 )  <_  B )
 )
 
Theoremnnaddm1cl 9260 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 )
 
Theoremnn0ltp1le 9261 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 ) )
 
Theoremnn0leltp1 9262 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremnn0ltlem1 9263 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
 ) ) )
 
Theoremznn0sub 9264 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9265.) (Contributed by NM, 14-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( N  -  M )  e.  NN0 ) )
 
Theoremnn0sub 9265 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )
 
Theoremltsubnn0 9266 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 ) )
 
Theoremnn0negleid 9267 A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
 |-  ( A  e.  NN0  ->  -u A  <_  A )
 
Theoremdifgtsumgt 9268 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 ) ) )
 
Theoremnn0n0n1ge2 9269 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 )
 
Theoremelz2 9270* 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 ) )
 
Theoremdfz2 9271 Alternate definition of the integers, based on elz2 9270. (Contributed by Mario Carneiro, 16-May-2014.)
 |- 
 ZZ  =  (  -  " ( NN  X.  NN ) )
 
Theoremnn0sub2 9272 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( N  -  M )  e. 
 NN0 )
 
Theoremzapne 9273 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 ) )
 
Theoremzdceq 9274 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  =  B )
 
Theoremzdcle 9275 Integer  <_ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <_  B )
 
Theoremzdclt 9276 Integer  < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
 
Theoremzltlen 9277 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8538 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 ) ) )
 
Theoremnn0n0n1ge2b 9278 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 )
 )
 
Theoremnn0lt10b 9279 A nonnegative integer less than  1 is  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  =  0
 ) )
 
Theoremnn0lt2 9280 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 ) )
 
Theoremnn0le2is012 9281 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 ) )
 
Theoremnn0lem1lt 9282 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
 
Theoremnnlem1lt 9283 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremnnltlem1 9284 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremnnm1ge0 9285 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 ) )
 
Theoremnn0ge0div 9286 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 ) )
 
Theoremzdiv 9287* 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 ) )
 
Theoremzdivadd 9288 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 )
 
Theoremzdivmul 9289 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 )
 
Theoremzextle 9290* 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 )
 
Theoremzextlt 9291* 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 )
 
Theoremrecnz 9292 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 )
 
Theorembtwnnz 9293 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 )
 
Theoremgtndiv 9294 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 )
 
Theoremhalfnz 9295 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 |- 
 -.  ( 1  / 
 2 )  e.  ZZ
 
Theorem3halfnz 9296 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
 |- 
 -.  ( 3  / 
 2 )  e.  ZZ
 
Theoremsuprzclex 9297* 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 )
 
Theoremprime 9298* 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 ) ) )
 
Theoremmsqznn 9299 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 )
 
Theoremzneo 9300 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 )
 )
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