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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0nn0 11901 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 0 ∈ ℕ0 | ||
Theorem | 1nn0 11902 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 1 ∈ ℕ0 | ||
Theorem | 2nn0 11903 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 2 ∈ ℕ0 | ||
Theorem | 3nn0 11904 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 3 ∈ ℕ0 | ||
Theorem | 4nn0 11905 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 4 ∈ ℕ0 | ||
Theorem | 5nn0 11906 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 5 ∈ ℕ0 | ||
Theorem | 6nn0 11907 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 6 ∈ ℕ0 | ||
Theorem | 7nn0 11908 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 7 ∈ ℕ0 | ||
Theorem | 8nn0 11909 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 8 ∈ ℕ0 | ||
Theorem | 9nn0 11910 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 9 ∈ ℕ0 | ||
Theorem | nn0ge0 11911 | A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | ||
Theorem | nn0nlt0 11912 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℕ0 → ¬ 𝐴 < 0) | ||
Theorem | nn0ge0i 11913 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 0 ≤ 𝑁 | ||
Theorem | nn0le0eq0 11914 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 ↔ 𝑁 = 0)) | ||
Theorem | nn0p1gt0 11915 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | ||
Theorem | nnnn0addcl 11916 | 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.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ) | ||
Theorem | nn0nnaddcl 11917 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | ||
Theorem | 0mnnnnn0 11918 | The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) | ||
Theorem | un0addcl 11919 | If 𝑆 is closed under addition, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ 𝑇 = (𝑆 ∪ {0}) & ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 + 𝑁) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 + 𝑁) ∈ 𝑇) | ||
Theorem | un0mulcl 11920 | If 𝑆 is closed under multiplication, then so is 𝑆 ∪ {0}. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ 𝑇 = (𝑆 ∪ {0}) & ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑆 ∧ 𝑁 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑀 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) → (𝑀 · 𝑁) ∈ 𝑇) | ||
Theorem | nn0addcl 11921 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0) | ||
Theorem | nn0mulcl 11922 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 · 𝑁) ∈ ℕ0) | ||
Theorem | nn0addcli 11923 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 + 𝑁) ∈ ℕ0 | ||
Theorem | nn0mulcli 11924 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑀 · 𝑁) ∈ ℕ0 | ||
Theorem | nn0p1nn 11925 | A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn 11639. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | ||
Theorem | peano2nn0 11926 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | ||
Theorem | nnm1nn0 11927 | A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | ||
Theorem | elnn0nn 11928 | The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℕ)) | ||
Theorem | elnnnn0 11929 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℂ ∧ (𝑁 − 1) ∈ ℕ0)) | ||
Theorem | elnnnn0b 11930 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | ||
Theorem | elnnnn0c 11931 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | ||
Theorem | nn0addge1 11932 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁)) | ||
Theorem | nn0addge2 11933 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝑁 + 𝐴)) | ||
Theorem | nn0addge1i 11934 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝐴 ≤ (𝐴 + 𝑁) | ||
Theorem | nn0addge2i 11935 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝐴 ≤ (𝑁 + 𝐴) | ||
Theorem | nn0sub 11936 | Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | ||
Theorem | ltsubnn0 11937 | 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐵 < 𝐴 → (𝐴 − 𝐵) ∈ ℕ0)) | ||
Theorem | nn0negleid 11938 | A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.) |
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴) | ||
Theorem | difgtsumgt 11939 | 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 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) | ||
Theorem | nn0le2xi 11940 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ≤ (2 · 𝑁) | ||
Theorem | nn0lele2xi 11941 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀)) | ||
Theorem | frnnn0supp 11942 | Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) | ||
Theorem | frnnn0fsupp 11943 | A function on ℕ0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.) |
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) | ||
Theorem | nnnn0d 11944 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0) | ||
Theorem | nn0red 11945 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | nn0cnd 11946 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | nn0ge0d 11947 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | nn0addcld 11948 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) | ||
Theorem | nn0mulcld 11949 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) | ||
Theorem | nn0readdcl 11950 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | nn0n0n1ge2 11951 | 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 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁) | ||
Theorem | nn0n0n1ge2b 11952 | 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 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | ||
Theorem | nn0ge2m1nn 11953 | 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.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) | ||
Theorem | nn0ge2m1nn0 11954 | 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.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ0) | ||
Theorem | nn0nndivcl 11955 | Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ) | ||
The function values of the hash (set size) function are either nonnegative integers or positive infinity, see hashf 13688. 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 ℝ*, see df-xr 10668. The definition of extended nonnegative integers can be used in Ramsey theory, because the Ramsey number is either a nonnegative integer or plus infinity, see ramcl2 16342, or for the degree of polynomials, see mdegcl 24592, or for the degree of vertices in graph theory, see vtxdgf 27181. | ||
Syntax | cxnn0 11956 | The set of extended nonnegative integers. |
class ℕ0* | ||
Definition | df-xnn0 11957 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers ℝ*, see df-xr 10668. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0* = (ℕ0 ∪ {+∞}) | ||
Theorem | elxnn0 11958 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | ||
Theorem | nn0ssxnn0 11959 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
⊢ ℕ0 ⊆ ℕ0* | ||
Theorem | nn0xnn0 11960 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*) | ||
Theorem | xnn0xr 11961 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | ||
Theorem | 0xnn0 11962 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ 0 ∈ ℕ0* | ||
Theorem | pnf0xnn0 11963 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
⊢ +∞ ∈ ℕ0* | ||
Theorem | nn0nepnf 11964 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0 → 𝐴 ≠ +∞) | ||
Theorem | nn0xnn0d 11965 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℕ0*) | ||
Theorem | nn0nepnfd 11966 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | xnn0nemnf 11967 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | ||
Theorem | xnn0xrnemnf 11968 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | ||
Theorem | xnn0nnn0pnf 11969 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) | ||
Syntax | cz 11970 | Extend class notation to include the class of integers. |
class ℤ | ||
Definition | df-z 11971 | 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.) |
⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)} | ||
Theorem | elz 11972 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | ||
Theorem | nnnegz 11973 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | ||
Theorem | zre 11974 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | ||
Theorem | zcn 11975 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | ||
Theorem | zrei 11976 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝐴 ∈ ℤ ⇒ ⊢ 𝐴 ∈ ℝ | ||
Theorem | zssre 11977 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℝ | ||
Theorem | zsscn 11978 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
⊢ ℤ ⊆ ℂ | ||
Theorem | zex 11979 | The set of integers exists. See also zexALT 11990. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ℤ ∈ V | ||
Theorem | elnnz 11980 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | ||
Theorem | 0z 11981 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
⊢ 0 ∈ ℤ | ||
Theorem | 0zd 11982 | Zero is an integer, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℤ) | ||
Theorem | elnn0z 11983 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) | ||
Theorem | elznn0nn 11984 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | ||
Theorem | elznn0 11985 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | elznn 11986 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ0))) | ||
Theorem | zle0orge1 11987 | There is no integer in the open unit interval, i.e., an integer is either less than or equal to 0 or greater than or equal to 1. (Contributed by AV, 4-Jun-2023.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ≤ 0 ∨ 1 ≤ 𝑍)) | ||
Theorem | elz2 11988* | 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 ∈ ℤ ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 − 𝑦)) | ||
Theorem | dfz2 11989 | Alternative definition of the integers, based on elz2 11988. (Contributed by Mario Carneiro, 16-May-2014.) |
⊢ ℤ = ( − “ (ℕ × ℕ)) | ||
Theorem | zexALT 11990 | Alternate proof of zex 11979. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℤ ∈ V | ||
Theorem | nnssz 11991 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022.) |
⊢ ℕ ⊆ ℤ | ||
Theorem | nn0ssz 11992 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
⊢ ℕ0 ⊆ ℤ | ||
Theorem | nnz 11993 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | ||
Theorem | nn0z 11994 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | ||
Theorem | nnzi 11995 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | nn0zi 11996 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ 𝑁 ∈ ℤ | ||
Theorem | elnnz1 11997 | Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | ||
Theorem | znnnlt1 11998 | An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.) |
⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1)) | ||
Theorem | nnzrab 11999 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥} | ||
Theorem | nn0zrab 12000 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥} |
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