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Theorem gtned 9709
Description: 'Less 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
gtned  |-  ( ph  ->  B  =/=  A )

Proof of Theorem gtned
StepHypRef Expression
1 ltd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 ltned.2 . 2  |-  ( ph  ->  A  <  B )
3 ltne 9670 . 2  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
41, 2, 3syl2anc 659 1  |-  ( ph  ->  B  =/=  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    =/= wne 2649   class class class wbr 4439   RRcr 9480    < clt 9617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622
This theorem is referenced by:  ltned  9710  seqf1olem1  12131  seqf1olem2  12132  hashfun  12482  abssubne0  13234  rpnnen2lem9  14043  rpnnen2lem11  14045  coe1tmmul2  18515  iccpnfcnv  21613  iccpnfhmeo  21614  pmltpclem2  22030  voliunlem1  22129  dvferm1lem  22554  lhop2  22585  ftc1lem5  22610  vieta1lem2  22876  geolim3  22904  logtayl  23212  atanre  23416  perfectlem2  23706  axlowdimlem16  24465  clwwisshclwwlem  25011  eupap1  25181  frgraogt3nreg  25325  nn0sqeq1  27796  esumcvgre  28323  eulerpartlems  28566  lgamgulmlem2  28839  lgamgulmlem3  28840  ivthALT  30396  pellfundne1  31067  eliccelioc  31803  fmul01lt1lem1  31820  lptre2pt  31888  cncfiooicclem1  31938  cncfioobdlem  31941  dvnmul  31982  ditgeqiooicc  32001  itgioocnicc  32018  iblcncfioo  32019  wallispilem4  32092  wallispi  32094  wallispi2lem1  32095  wallispi2lem2  32096  wallispi2  32097  stirlinglem5  32102  fourierdlem4  32135  fourierdlem34  32165  fourierdlem41  32172  fourierdlem42  32173  fourierdlem48  32179  fourierdlem49  32180  fourierdlem61  32192  fourierdlem73  32204  fourierdlem75  32206  fourierdlem76  32207  fourierdlem81  32212  fourierdlem82  32213  fourierdlem84  32215  fourierdlem93  32224  fourierdlem111  32242  fouriersw  32256  etransclem35  32294  perfectALTVlem2  32616
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