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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0le0 8901 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 8902 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 8903 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 8904 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem2ap0 8905 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0

Theorem3pos 8906 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 8907 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem3ap0 8908 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0

Theorem4pos 8909 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 8910 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem4ap0 8911 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0

Theorem5pos 8912 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 8913 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 8914 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 8915 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 8916 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

4.4.4  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 8917 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 8918 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 8919 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 8920 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremneg1ap0 8921 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0

Theoremnegneg1e1 8922 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 8923 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 8924 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 8925 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 8926 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theoremfv0p1e1 8927 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))

Theorem1p0e1 8928 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 8929 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 8930 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8957. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 8931 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 8932 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2

Theorem4m1e3 8933 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
(4 − 1) = 3

Theorem5m1e4 8934 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
(5 − 1) = 4

Theorem6m1e5 8935 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5

Theorem7m1e6 8936 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6

Theorem8m1e7 8937 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7

Theorem9m1e8 8938 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8

Theorem2p2e4 8939 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4

Theorem2times 8940 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 8941 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 8942 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 8943 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

Theorem2div2e1 8944 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

Theorem2p1e3 8945 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3

Theorem1p2e3 8946 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3

Theorem3p1e4 8947 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4

Theorem4p1e5 8948 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5

Theorem5p1e6 8949 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6

Theorem6p1e7 8950 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7

Theorem7p1e8 8951 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8

Theorem8p1e9 8952 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9

Theorem3p2e5 8953 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5

Theorem3p3e6 8954 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6

Theorem4p2e6 8955 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6

Theorem4p3e7 8956 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7

Theorem4p4e8 8957 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8

Theorem5p2e7 8958 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7

Theorem5p3e8 8959 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8

Theorem5p4e9 8960 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem6p2e8 8961 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 8962 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem7p2e9 8963 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem1t1e1 8964 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 8965 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

Theorem2t2e4 8966 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4

Theorem3t1e3 8967 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3

Theorem3t2e6 8968 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6

Theorem3t3e9 8969 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9

Theorem4t2e8 8970 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8

Theorem2t0e0 8971 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0

Theorem4d2e2 8972 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2

Theorem2nn 8973 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem3nn 8974 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem4nn 8975 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem5nn 8976 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem6nn 8977 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem7nn 8978 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem8nn 8979 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem9nn 8980 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem1lt2 8981 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2

Theorem2lt3 8982 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3

Theorem1lt3 8983 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3

Theorem3lt4 8984 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4

Theorem2lt4 8985 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4

Theorem1lt4 8986 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 8987 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 8988 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 8989 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 8990 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 8991 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 8992 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 8993 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 8994 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 8995 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 8996 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 8997 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 8998 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 8999 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 9000 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

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