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Theorem ltnlei 8939
Description: 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)
Hypotheses
Ref Expression
lt.1  |-  A  e.  RR
lt.2  |-  B  e.  RR
Assertion
Ref Expression
ltnlei  |-  ( A  <  B  <->  -.  B  <_  A )

Proof of Theorem ltnlei
StepHypRef Expression
1 lt.2 . . 3  |-  B  e.  RR
2 lt.1 . . 3  |-  A  e.  RR
31, 2lenlti 8938 . 2  |-  ( B  <_  A  <->  -.  A  <  B )
43con2bii 322 1  |-  ( A  <  B  <->  -.  B  <_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    e. wcel 1684   class class class wbr 4023   RRcr 8736    < clt 8867    <_ cle 8868
This theorem is referenced by:  letrii  8944  divalglem5  12596  divalglem6  12597  bitsfzolem  12625  bitsfzo  12626  bitsinv1lem  12632  sadcadd  12649  strlemor1  13235  htpycc  18478  pco1  18513  pcohtpylem  18517  pcopt  18520  pcopt2  18521  pcoass  18522  pcorevlem  18524  vitalilem5  18967  vieta1lem2  19691  ppiltx  20415  ppiublem1  20441  chtub  20451  ballotlem2  23047  rnlogblem  23401  subfacp1lem1  23710  subfacp1lem5  23715  axlowdimlem16  24585  axlowdim  24589  fdc  26455  pellexlem6  26919  jm2.23  27089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-xr 8871  df-le 8873
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