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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxc4 8401 Extend class notation to include the number 4.
class 4
 
Syntaxc5 8402 Extend class notation to include the number 5.
class 5
 
Syntaxc6 8403 Extend class notation to include the number 6.
class 6
 
Syntaxc7 8404 Extend class notation to include the number 7.
class 7
 
Syntaxc8 8405 Extend class notation to include the number 8.
class 8
 
Syntaxc9 8406 Extend class notation to include the number 9.
class 9
 
Syntaxc10 8407 Extend class notation to include the number 10.
class 10
 
Definitiondf-2 8408 Define the number 2. (Contributed by NM, 27-May-1999.)
2 = (1 + 1)
 
Definitiondf-3 8409 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)
 
Definitiondf-4 8410 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)
 
Definitiondf-5 8411 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)
 
Definitiondf-6 8412 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)
 
Definitiondf-7 8413 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)
 
Definitiondf-8 8414 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)
 
Definitiondf-9 8415 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)
 
Theorem0ne1 8416 0 ≠ 1 (common case). See aso 1ap0 8000. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1
 
Theorem1ne0 8417 1 ≠ 0. See aso 1ap0 8000. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0
 
Theorem1m1e0 8418 (1 − 1) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0
 
Theorem2re 8419 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ
 
Theorem2cn 8420 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
2 ∈ ℂ
 
Theorem2ex 8421 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V
 
Theorem2cnd 8422 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)
 
Theorem3re 8423 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ
 
Theorem3cn 8424 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
3 ∈ ℂ
 
Theorem3ex 8425 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V
 
Theorem4re 8426 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ
 
Theorem4cn 8427 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
4 ∈ ℂ
 
Theorem5re 8428 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ
 
Theorem5cn 8429 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
5 ∈ ℂ
 
Theorem6re 8430 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ
 
Theorem6cn 8431 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
6 ∈ ℂ
 
Theorem7re 8432 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ
 
Theorem7cn 8433 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
7 ∈ ℂ
 
Theorem8re 8434 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ
 
Theorem8cn 8435 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
8 ∈ ℂ
 
Theorem9re 8436 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ
 
Theorem9cn 8437 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
9 ∈ ℂ
 
Theorem0le0 8438 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0
 
Theorem0le2 8439 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2
 
Theorem2pos 8440 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2
 
Theorem2ne0 8441 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0
 
Theorem2ap0 8442 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0
 
Theorem3pos 8443 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3
 
Theorem3ne0 8444 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0
 
Theorem3ap0 8445 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0
 
Theorem4pos 8446 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4
 
Theorem4ne0 8447 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0
 
Theorem4ap0 8448 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0
 
Theorem5pos 8449 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5
 
Theorem6pos 8450 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6
 
Theorem7pos 8451 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7
 
Theorem8pos 8452 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8
 
Theorem9pos 8453 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9
 
3.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

 
Theoremneg1cn 8454 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ
 
Theoremneg1rr 8455 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ
 
Theoremneg1ne0 8456 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0
 
Theoremneg1lt0 8457 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0
 
Theoremneg1ap0 8458 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0
 
Theoremnegneg1e1 8459 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1
 
Theorem1pneg1e0 8460 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0
 
Theorem0m0e0 8461 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0
 
Theorem1m0e1 8462 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1
 
Theorem0p1e1 8463 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1
 
Theorem1p0e1 8464 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1
 
Theorem1p1e2 8465 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2
 
Theorem2m1e1 8466 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8487. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1
 
Theorem1e2m1 8467 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)
 
Theorem3m1e2 8468 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2
 
Theorem2p2e4 8469 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4
 
Theorem2times 8470 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))
 
Theoremtimes2 8471 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))
 
Theorem2timesi 8472 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)
 
Theoremtimes2i 8473 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)
 
Theorem2div2e1 8474 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1
 
Theorem2p1e3 8475 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3
 
Theorem1p2e3 8476 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3
 
Theorem3p1e4 8477 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4
 
Theorem4p1e5 8478 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5
 
Theorem5p1e6 8479 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6
 
Theorem6p1e7 8480 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7
 
Theorem7p1e8 8481 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8
 
Theorem8p1e9 8482 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9
 
Theorem3p2e5 8483 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5
 
Theorem3p3e6 8484 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6
 
Theorem4p2e6 8485 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6
 
Theorem4p3e7 8486 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7
 
Theorem4p4e8 8487 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8
 
Theorem5p2e7 8488 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7
 
Theorem5p3e8 8489 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8
 
Theorem5p4e9 8490 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9
 
Theorem6p2e8 8491 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8
 
Theorem6p3e9 8492 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9
 
Theorem7p2e9 8493 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9
 
Theorem1t1e1 8494 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1
 
Theorem2t1e2 8495 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2
 
Theorem2t2e4 8496 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4
 
Theorem3t1e3 8497 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3
 
Theorem3t2e6 8498 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6
 
Theorem3t3e9 8499 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9
 
Theorem4t2e8 8500 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8
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