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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nn0rei 9101 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
Theorem | nn0cni 9102 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
Theorem | dfn2 9103 | The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
Theorem | elnnne0 9104 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Theorem | 0nn0 9105 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 1nn0 9106 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 2nn0 9107 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 3nn0 9108 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 4nn0 9109 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 5nn0 9110 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 6nn0 9111 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 7nn0 9112 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 8nn0 9113 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 9nn0 9114 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | nn0ge0 9115 | A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nn0nlt0 9116 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0i 9117 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0le0eq0 9118 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
Theorem | nn0p1gt0 9119 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Theorem | nnnn0addcl 9120 | A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nn0nnaddcl 9121 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 9122 | The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
Theorem | un0addcl 9123 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 9124 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 9125 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 9126 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 9127 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 9128 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 9129 | A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | peano2nn0 9130 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 9131 | A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | elnn0nn 9132 | The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | elnnnn0 9133 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 9134 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 9135 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 9136 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 9137 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 9138 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 9139 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 9140 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 9141 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 9142 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 9143 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 9144 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 9145 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 9146 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 9147 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 9148 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 9149 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 9150 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
Theorem | nn0ge2m1nn0 9151 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
Theorem | nn0nndivcl 9152 | Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers , see df-xr 7916. | ||
Syntax | cxnn0 9153 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 9154 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7916. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7064 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 9155 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0ssxnn0 9156 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0xnn0 9157 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xr 9158 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | 0xnn0 9159 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | pnf0xnn0 9160 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnf 9161 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Theorem | nn0xnn0d 9162 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnfd 9163 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
Theorem | xnn0nemnf 9164 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xrnemnf 9165 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0nnn0pnf 9166 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Syntax | cz 9167 | Extend class notation to include the class of integers. |
Definition | df-z 9168 | Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.) |
Theorem | elz 9169 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | nnnegz 9170 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | zre 9171 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
Theorem | zcn 9172 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | zrei 9173 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
Theorem | zssre 9174 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
Theorem | zsscn 9175 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | zex 9176 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | elnnz 9177 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | 0z 9178 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | 0zd 9179 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elnn0z 9180 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn0nn 9181 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Theorem | elznn0 9182 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn 9183 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
Theorem | nnssz 9184 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
Theorem | nn0ssz 9185 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnz 9186 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nn0z 9187 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnzi 9188 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nn0zi 9189 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | elnnz1 9190 | Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nnzrab 9191 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | nn0zrab 9192 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | 1z 9193 | One is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 1zzd 9194 | 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | 2z 9195 | Two is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 3z 9196 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4z 9197 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
Theorem | znegcl 9198 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
Theorem | neg1z 9199 | -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
Theorem | znegclb 9200 | A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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