P. 100 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	8t3e24 9901	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 3) = ;24
Theorem	8t4e32 9902	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 4) = ;32
Theorem	8t5e40 9903	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 5) = ;40
Theorem	8t6e48 9904	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 6) = ;48
Theorem	8t7e56 9905	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 7) = ;56
Theorem	8t8e64 9906	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (8 · 8) = ;64
Theorem	9t2e18 9907	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 2) = ;18
Theorem	9t3e27 9908	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 3) = ;27
Theorem	9t4e36 9909	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 4) = ;36
Theorem	9t5e45 9910	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 5) = ;45
Theorem	9t6e54 9911	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 6) = ;54
Theorem	9t7e63 9912	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 7) = ;63
Theorem	9t8e72 9913	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 8) = ;72
Theorem	9t9e81 9914	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ (9 · 9) = ;81
Theorem	9t11e99 9915	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ (9 · ;11) = ;99
Theorem	9lt10 9916	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10
Theorem	8lt10 9917	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 8 < ;10
Theorem	7lt10 9918	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 7 < ;10
Theorem	6lt10 9919	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 6 < ;10
Theorem	5lt10 9920	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 5 < ;10
Theorem	4lt10 9921	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 4 < ;10
Theorem	3lt10 9922	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 3 < ;10
Theorem	2lt10 9923	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 2 < ;10
Theorem	1lt10 9924	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 1 < ;10
Theorem	decbin0 9925	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (4 · 𝐴) = (2 · (2 · 𝐴))
Theorem	decbin2 9926	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
Theorem	decbin3 9927	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)
Theorem	halfthird 9928	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
Theorem	5recm6rec 9929	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30)
4.4.11  Upper sets of integers
Syntax	cuz 9930	Extend class notation with the upper integer function. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀".
class ℤ≥
Definition	df-uz 9931*	Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀". See uzval 9932 for its value, uzssz 9951 for its relationship to ℤ, nnuz 9967 and nn0uz 9966 for its relationships to ℕ and ℕ0, and eluz1 9934 and eluz2 9936 for its membership relations. (Contributed by NM, 5-Sep-2005.)
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
Theorem	uzval 9932*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})
Theorem	uzf 9933	The domain and codomain of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ℤ≥:ℤ⟶𝒫 ℤ
Theorem	eluz1 9934	Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))
Theorem	eluzel2 9935	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
Theorem	eluz2 9936	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 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzmn 9937	Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ (ℤ≥‘(𝑀 − 𝑁)))
Theorem	eluz1i 9938	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
⊢ 𝑀 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
Theorem	eluzuzle 9939	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 ∈ ℤ ∧ 𝐵 ≤ 𝐴) → (𝐶 ∈ (ℤ≥‘𝐴) → 𝐶 ∈ (ℤ≥‘𝐵)))
Theorem	eluzelz 9940	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
Theorem	eluzelre 9941	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ)
Theorem	eluzelcn 9942	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ)
Theorem	eluzle 9943	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁)
Theorem	eluz 9944	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
Theorem	uzid 9945	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzidd 9946	Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
Theorem	uzn0 9947	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅)
Theorem	uztrn 9948	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁))
Theorem	uztrn2 9949	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝐾)    ⇒   ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍)
Theorem	uzneg 9950	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁))
Theorem	uzssz 9951	An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (ℤ≥‘𝑀) ⊆ ℤ
Theorem	uzss 9952	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀))
Theorem	uztric 9953	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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁)))
Theorem	uz11 9954	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁))
Theorem	eluzp1m1 9955	Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀))
Theorem	eluzp1l 9956	Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁)
Theorem	eluzp1p1 9957	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
Theorem	eluzaddi 9958	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsubi 9959	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝐾 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	eluzadd 9960	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
Theorem	eluzsub 9961	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	uzm1 9962	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ≥‘𝑀)))
Theorem	uznn0sub 9963	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	uzin 9964	Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ≥‘𝑀) ∩ (ℤ≥‘𝑁)) = (ℤ≥‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
Theorem	uzp1 9965	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
Theorem	nn0uz 9966	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ0 = (ℤ≥‘0)
Theorem	nnuz 9967	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
⊢ ℕ = (ℤ≥‘1)
Theorem	elnnuz 9968	A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
Theorem	elnn0uz 9969	A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
Theorem	5eluz3 9970	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
⊢ 5 ∈ (ℤ≥‘3)
Theorem	uzuzle23 9971	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 ∈ (ℤ≥‘3) → 𝐴 ∈ (ℤ≥‘2))
Theorem	uzuzle24 9972	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘2))
Theorem	uzuzle34 9973	An integer greater than or equal to 4 is an integer greater than or equal to 3. (Contributed by AV, 5-Sep-2025.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘3))
Theorem	uzuzle35 9974	An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
⊢ (𝐴 ∈ (ℤ≥‘5) → 𝐴 ∈ (ℤ≥‘3))
Theorem	eluz2nn 9975	An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ)
Theorem	eluz3nn 9976	An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
Theorem	eluz4eluz2 9977	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘2))
Theorem	eluz4nn 9978	An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ ℕ)
Theorem	eluzge2nn0 9979	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 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ0)
Theorem	eluz2n0 9980	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0)
Theorem	eluzge3nn 9981	If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ)
Theorem	uz3m2nn 9982	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 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ)
Theorem	1eluzge0 9983	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 1 ∈ (ℤ≥‘0)
Theorem	2eluzge0 9984	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 ∈ (ℤ≥‘0)
Theorem	2eluzge1 9985	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 2 ∈ (ℤ≥‘1)
Theorem	uznnssnn 9986	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
⊢ (𝑁 ∈ ℕ → (ℤ≥‘𝑁) ⊆ ℕ)
Theorem	raluz 9987*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	raluz2 9988*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	rexuz 9989*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	rexuz2 9990*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	2rexuz 9991*	Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))
Theorem	peano2uz 9992	Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
Theorem	peano2uzs 9993	Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍)
Theorem	peano2uzr 9994	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	uzaddcl 9995	Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	nn0pzuz 9996	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 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍))
Theorem	uzind4 9997*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4ALT 9998*	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 9997 or uzind4ALT 9998 may be used; see comment for nnind 9322. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    &   ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4s 9999*	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 ∈ ℤ → [𝑀 / 𝑘]𝜑)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑)
Theorem	uzind4s2 10000*	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 9999 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑)