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Theorem ltnlei 11367
Description: 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)
Hypotheses
Ref Expression
lt.1 𝐴 ∈ ℝ
lt.2 𝐵 ∈ ℝ
Assertion
Ref Expression
ltnlei (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)

Proof of Theorem ltnlei
StepHypRef Expression
1 lt.2 . . 3 𝐵 ∈ ℝ
2 lt.1 . . 3 𝐴 ∈ ℝ
31, 2lenlti 11366 . 2 (𝐵𝐴 ↔ ¬ 𝐴 < 𝐵)
43con2bii 356 1 (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2098   class class class wbr 5149  cr 11139   < clt 11280  cle 11281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-xr 11284  df-le 11286
This theorem is referenced by:  letrii  11371  nn0ge2m1nn  12574  0nelfz1  13555  fzpreddisj  13585  hashnn0n0nn  14386  hashge2el2dif  14477  divalglem5  16377  divalglem6  16378  sadcadd  16436  htpycc  24950  pco1  24986  pcohtpylem  24990  pcopt  24993  pcopt2  24994  pcoass  24995  pcorevlem  24997  vitalilem5  25585  vieta1lem2  26291  ppiltx  27154  ppiublem1  27180  chtub  27190  axlowdimlem16  28840  axlowdim  28844  lfgrnloop  29010  lfuhgr1v0e  29139  lfgrwlkprop  29573  ballotlem2  34239  subfacp1lem1  34920  subfacp1lem5  34925  bcneg1  35461  poimirlem9  37233  poimirlem16  37240  poimirlem17  37241  poimirlem19  37243  poimirlem20  37244  poimirlem22  37246  fdc  37349  pellexlem6  42396  jm2.23  42559
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