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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremneg1rr 7801 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 -u 1  RR
 
Theoremneg1ne0 7802 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u 1  =/=  0
 
Theoremneg1lt0 7803 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u 1  <  0
 
Theoremneg1ap0 7804 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 -u 1 #  0
 
Theoremnegneg1e1 7805  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u -u 1  1
 
Theorem1pneg1e0 7806  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  -u 1  0
 
Theorem0m0e0 7807 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  -  0  0
 
Theorem1m0e1 7808 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  -  0  1
 
Theorem0p1e1 7809 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 0  +  1  1
 
Theorem1p0e1 7810 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  0  1
 
Theorem1p1e2 7811 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 1  +  1  2
 
Theorem2m1e1 7812 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 7834. (Contributed by David A. Wheeler, 4-Jan-2017.)
 2  -  1  1
 
Theorem1e2m1 7813 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  2  -  1
 
Theorem3m1e2 7814 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 3  -  1  2
 
Theorem2p2e4 7815 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 7816 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
 CC  2  x.  +
 
Theoremtimes2 7817 A number times 2. (Contributed by NM, 16-Oct-2007.)
 CC  x.  2  +
 
Theorem2timesi 7818 Two times a number. (Contributed by NM, 1-Aug-1999.)
 CC   =>     2  x.  +
 
Theoremtimes2i 7819 A number times 2. (Contributed by NM, 11-May-2004.)
 CC   =>     x.  2  +
 
Theorem2div2e1 7820 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  2  1
 
Theorem2p1e3 7821 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 2  +  1  3
 
Theorem1p2e3 7822 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  2  3
 
Theorem3p1e4 7823 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 3  +  1  4
 
Theorem4p1e5 7824 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 4  +  1  5
 
Theorem5p1e6 7825 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 5  +  1  6
 
Theorem6p1e7 7826 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 6  +  1  7
 
Theorem7p1e8 7827 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 7  +  1  8
 
Theorem8p1e9 7828 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 8  +  1  9
 
Theorem9p1e10 7829 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
 9  +  1  10
 
Theorem3p2e5 7830 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 3  +  2  5
 
Theorem3p3e6 7831 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 3  +  3  6
 
Theorem4p2e6 7832 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 4  +  2  6
 
Theorem4p3e7 7833 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 4  +  3  7
 
Theorem4p4e8 7834 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 4  +  4  8
 
Theorem5p2e7 7835 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 5  +  2  7
 
Theorem5p3e8 7836 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 5  +  3  8
 
Theorem5p4e9 7837 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 5  +  4  9
 
Theorem5p5e10 7838 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
 5  +  5  10
 
Theorem6p2e8 7839 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 6  +  2  8
 
Theorem6p3e9 7840 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 6  +  3  9
 
Theorem6p4e10 7841 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
 6  +  4  10
 
Theorem7p2e9 7842 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 7  +  2  9
 
Theorem7p3e10 7843 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
 7  +  3  10
 
Theorem8p2e10 7844 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
 8  +  2  10
 
Theorem1t1e1 7845 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 1  x.  1  1
 
Theorem2t1e2 7846 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
 2  x.  1  2
 
Theorem2t2e4 7847 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 2  x.  2  4
 
Theorem3t1e3 7848 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 3  x.  1  3
 
Theorem3t2e6 7849 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 3  x.  2  6
 
Theorem3t3e9 7850 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 3  x.  3  9
 
Theorem4t2e8 7851 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 4  x.  2  8
 
Theorem5t2e10 7852 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
 5  x.  2  10
 
Theorem2t0e0 7853 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  x.  0  0
 
Theorem4d2e2 7854 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 4  2  2
 
Theorem2nn 7855 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
 2  NN
 
Theorem3nn 7856 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 3  NN
 
Theorem4nn 7857 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 4  NN
 
Theorem5nn 7858 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 5  NN
 
Theorem6nn 7859 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 6  NN
 
Theorem7nn 7860 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 7  NN
 
Theorem8nn 7861 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 8  NN
 
Theorem9nn 7862 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
 9  NN
 
Theorem10nn 7863 10 is a positive integer. (Contributed by NM, 8-Nov-2012.)

 10  NN
 
Theorem1lt2 7864 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 1  <  2
 
Theorem2lt3 7865 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 2  <  3
 
Theorem1lt3 7866 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 1  <  3
 
Theorem3lt4 7867 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 3  <  4
 
Theorem2lt4 7868 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 2  <  4
 
Theorem1lt4 7869 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 1  <  4
 
Theorem4lt5 7870 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 4  <  5
 
Theorem3lt5 7871 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 3  <  5
 
Theorem2lt5 7872 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 2  <  5
 
Theorem1lt5 7873 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 1  <  5
 
Theorem5lt6 7874 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 5  <  6
 
Theorem4lt6 7875 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 4  <  6
 
Theorem3lt6 7876 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 3  <  6
 
Theorem2lt6 7877 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 2  <  6
 
Theorem1lt6 7878 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 1  <  6
 
Theorem6lt7 7879 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 6  <  7
 
Theorem5lt7 7880 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 5  <  7
 
Theorem4lt7 7881 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 4  <  7
 
Theorem3lt7 7882 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 3  <  7
 
Theorem2lt7 7883 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 2  <  7
 
Theorem1lt7 7884 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 1  <  7
 
Theorem7lt8 7885 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 7  <  8
 
Theorem6lt8 7886 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 6  <  8
 
Theorem5lt8 7887 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 5  <  8
 
Theorem4lt8 7888 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 4  <  8
 
Theorem3lt8 7889 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 3  <  8
 
Theorem2lt8 7890 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 2  <  8
 
Theorem1lt8 7891 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 1  <  8
 
Theorem8lt9 7892 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 8  <  9
 
Theorem7lt9 7893 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 7  <  9
 
Theorem6lt9 7894 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 6  <  9
 
Theorem5lt9 7895 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 5  <  9
 
Theorem4lt9 7896 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 4  <  9
 
Theorem3lt9 7897 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 3  <  9
 
Theorem2lt9 7898 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 2  <  9
 
Theorem1lt9 7899 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 1  <  9
 
Theorem9lt10 7900 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 9  <  10
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