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Theorem ltned 8066
Description: 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
ltd.1  |-  ( ph  ->  A  e.  RR )
ltned.2  |-  ( ph  ->  A  <  B )
Assertion
Ref Expression
ltned  |-  ( ph  ->  A  =/=  B )

Proof of Theorem ltned
StepHypRef Expression
1 ltd.1 . . 3  |-  ( ph  ->  A  e.  RR )
2 ltned.2 . . 3  |-  ( ph  ->  A  <  B )
31, 2gtned 8065 . 2  |-  ( ph  ->  B  =/=  A )
43necomd 2433 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148    =/= wne 2347   class class class wbr 4002   RRcr 7806    < clt 7987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-pre-ltirr 7919
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-pnf 7989  df-mnf 7990  df-ltxr 7992
This theorem is referenced by:  modsumfzodifsn  10390  nprm  12114  4sqlem10  12376  trilpolemeq1  14639
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