P. 124 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	c9 12301	Extend class notation to include the number 9.
class 9
Definition	df-2 12302	Define the number 2. (Contributed by NM, 27-May-1999.)
⊢ 2 = (1 + 1)
Definition	df-3 12303	Define the number 3. (Contributed by NM, 27-May-1999.)
⊢ 3 = (2 + 1)
Definition	df-4 12304	Define the number 4. (Contributed by NM, 27-May-1999.)
⊢ 4 = (3 + 1)
Definition	df-5 12305	Define the number 5. (Contributed by NM, 27-May-1999.)
⊢ 5 = (4 + 1)
Definition	df-6 12306	Define the number 6. (Contributed by NM, 27-May-1999.)
⊢ 6 = (5 + 1)
Definition	df-7 12307	Define the number 7. (Contributed by NM, 27-May-1999.)
⊢ 7 = (6 + 1)
Definition	df-8 12308	Define the number 8. (Contributed by NM, 27-May-1999.)
⊢ 8 = (7 + 1)
Definition	df-9 12309	Define the number 9. (Contributed by NM, 27-May-1999.)
⊢ 9 = (8 + 1)
Theorem	1eltp012 12310	1 is an element of {0, 1, 2}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1, 2}
Theorem	0ne1 12311	Zero is different from one (the commuted form is Axiom ax-1ne0 11168). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≠ 1
Theorem	1m1e0 12312	One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (1 − 1) = 0
Theorem	2nn 12313	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
⊢ 2 ∈ ℕ
Theorem	2re 12314	The number 2 is real. (Contributed by NM, 27-May-1999.)
⊢ 2 ∈ ℝ
Theorem	2cn 12315	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 2 ∈ ℂ
Theorem	2cnALT 12316	Alternate proof of 2cn 12315. Shorter but uses more axioms. Similar proofs are possible for 3cn 12321, ... , 9cn 12340. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2 ∈ ℂ
Theorem	2ex 12317	The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 2 ∈ V
Theorem	2cnd 12318	The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 2 ∈ ℂ)
Theorem	3nn 12319	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 3 ∈ ℕ
Theorem	3re 12320	The number 3 is real. (Contributed by NM, 27-May-1999.)
⊢ 3 ∈ ℝ
Theorem	3cn 12321	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 3 ∈ ℂ
Theorem	3ex 12322	The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ V
Theorem	4nn 12323	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
⊢ 4 ∈ ℕ
Theorem	4re 12324	The number 4 is real. (Contributed by NM, 27-May-1999.)
⊢ 4 ∈ ℝ
Theorem	4cn 12325	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 4 ∈ ℂ
Theorem	5nn 12326	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 5 ∈ ℕ
Theorem	5re 12327	The number 5 is real. (Contributed by NM, 27-May-1999.)
⊢ 5 ∈ ℝ
Theorem	5cn 12328	The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 5 ∈ ℂ
Theorem	6nn 12329	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 6 ∈ ℕ
Theorem	6re 12330	The number 6 is real. (Contributed by NM, 27-May-1999.)
⊢ 6 ∈ ℝ
Theorem	6cn 12331	The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 6 ∈ ℂ
Theorem	7nn 12332	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 7 ∈ ℕ
Theorem	7re 12333	The number 7 is real. (Contributed by NM, 27-May-1999.)
⊢ 7 ∈ ℝ
Theorem	7cn 12334	The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 7 ∈ ℂ
Theorem	8nn 12335	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ 8 ∈ ℕ
Theorem	8re 12336	The number 8 is real. (Contributed by NM, 27-May-1999.)
⊢ 8 ∈ ℝ
Theorem	8cn 12337	The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 8 ∈ ℂ
Theorem	9nn 12338	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
⊢ 9 ∈ ℕ
Theorem	9re 12339	The number 9 is real. (Contributed by NM, 27-May-1999.)
⊢ 9 ∈ ℝ
Theorem	9cn 12340	The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 9 ∈ ℂ
Theorem	0le0 12341	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ≤ 0
Theorem	0le2 12342	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ≤ 2
Theorem	0le2OLD 12343	Obsolete version of 0le2 12342 as of 10-Jun-2026. (Contributed by David A. Wheeler, 7-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ≤ 2
Theorem	2pos 12344	The number 2 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 2
Theorem	2posOLD 12345	Obsolete version of 2pos 12344 as of 10-Jun-2026. (Contributed by NM, 27-May-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 < 2
Theorem	2ne0 12346	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
⊢ 2 ≠ 0
Theorem	2thalfe1 12347	2 times one half equals 1. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · (1 / 2)) = 1
Theorem	3pos 12348	The number 3 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 3
Theorem	3ne0 12349	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 3 ≠ 0
Theorem	4pos 12350	The number 4 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 4
Theorem	4ne0 12351	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ 4 ≠ 0
Theorem	5pos 12352	The number 5 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 5
Theorem	6pos 12353	The number 6 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 6
Theorem	7pos 12354	The number 7 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 7
Theorem	8pos 12355	The number 8 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 8
Theorem	9pos 12356	The number 9 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 9
5.4.4  Some properties of specific numbers This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering).
Theorem	1pneg1e0 12357	1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + -1) = 0
Theorem	0m0e0 12358	0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0 − 0) = 0
Theorem	1m0e1 12359	1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 − 0) = 1
Theorem	0p1e1 12360	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ (0 + 1) = 1
Theorem	fv0p1e1 12361	Function value at 𝑁 + 1 with 𝑁 replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
⊢ (𝑁 = 0 → (𝐹‘(𝑁 + 1)) = (𝐹‘1))
Theorem	1p0e1 12362	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (1 + 0) = 1
Theorem	1p1e2 12363	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
⊢ (1 + 1) = 2
Theorem	2m1e1 12364	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12394. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 − 1) = 1
Theorem	2m1e1OLD 12365	Obsolete version of 2m1e1 12364 as of 10-Jun-2026. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (2 − 1) = 1
Theorem	1e2m1 12366	1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 1 = (2 − 1)
Theorem	3m1e2 12367	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (3 − 1) = 2
Theorem	4m1e3 12368	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
⊢ (4 − 1) = 3
Theorem	5m1e4 12369	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
⊢ (5 − 1) = 4
Theorem	6m1e5 12370	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
⊢ (6 − 1) = 5
Theorem	7m1e6 12371	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
⊢ (7 − 1) = 6
Theorem	8m1e7 12372	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
⊢ (8 − 1) = 7
Theorem	9m1e8 12373	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
⊢ (9 − 1) = 8
Theorem	2p2e4 12374	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 8525 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.)
⊢ (2 + 2) = 4
Theorem	2times 12375	Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))
Theorem	times2 12376	A number times 2. (Contributed by NM, 16-Oct-2007.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
Theorem	2timesi 12377	Two times a number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (2 · 𝐴) = (𝐴 + 𝐴)
Theorem	times2i 12378	A number times 2. (Contributed by NM, 11-May-2004.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 2) = (𝐴 + 𝐴)
Theorem	2txmxeqx 12379	Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋)
Theorem	2div2e1 12380	2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (2 / 2) = 1
Theorem	2p1e3 12381	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (2 + 1) = 3
Theorem	1p2e3 12382	1 + 2 = 3. For a shorter proof using addcomli 11401, see 1p2e3ALT 12383. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
⊢ (1 + 2) = 3
Theorem	1p2e3ALT 12383	Alternate proof of 1p2e3 12382, shorter but using more axioms. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (1 + 2) = 3
Theorem	3p1e4 12384	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (3 + 1) = 4
Theorem	4p1e5 12385	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (4 + 1) = 5
Theorem	5p1e6 12386	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (5 + 1) = 6
Theorem	6p1e7 12387	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (6 + 1) = 7
Theorem	7p1e8 12388	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (7 + 1) = 8
Theorem	8p1e9 12389	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
⊢ (8 + 1) = 9
Theorem	3p2e5 12390	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
⊢ (3 + 2) = 5
Theorem	3p3e6 12391	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
⊢ (3 + 3) = 6
Theorem	4p2e6 12392	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
⊢ (4 + 2) = 6
Theorem	4p3e7 12393	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
⊢ (4 + 3) = 7
Theorem	4p4e8 12394	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
⊢ (4 + 4) = 8
Theorem	5p2e7 12395	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
⊢ (5 + 2) = 7
Theorem	5p3e8 12396	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
⊢ (5 + 3) = 8
Theorem	5p4e9 12397	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
⊢ (5 + 4) = 9
Theorem	6p2e8 12398	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
⊢ (6 + 2) = 8
Theorem	6p3e9 12399	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
⊢ (6 + 3) = 9
Theorem	7p2e9 12400	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
⊢ (7 + 2) = 9