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Theorem ltnlei 9701
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 9700 . 2  |-  ( B  <_  A  <->  -.  A  <  B )
43con2bii 332 1  |-  ( A  <  B  <->  -.  B  <_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1767   class class class wbr 4447   RRcr 9487    < clt 9624    <_ cle 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-xr 9628  df-le 9630
This theorem is referenced by:  letrii  9705  nn0ge2m1nn  10857  fzpreddisj  11725  hashnn0n0nn  12422  hashge2el2dif  12483  n2dvds1  13890  divalglem5  13910  divalglem6  13911  sadcadd  13963  strlemor1  14578  htpycc  21215  pco1  21250  pcohtpylem  21254  pcopt  21257  pcopt2  21258  pcoass  21259  pcorevlem  21261  vitalilem5  21756  vieta1lem2  22441  ppiltx  23179  ppiublem1  23205  chtub  23215  axlowdimlem16  23936  axlowdim  23940  spthispth  24251  rnlogblem  27655  ballotlem2  28067  subfacp1lem1  28263  subfacp1lem5  28268  fdc  29841  pellexlem6  30374  jm2.23  30542
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