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Theorem ltnlei 11335
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 11334 . 2 (𝐵𝐴 ↔ ¬ 𝐴 < 𝐵)
43con2bii 358 1 (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2107   class class class wbr 5149  cr 11109   < clt 11248  cle 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-xr 11252  df-le 11254
This theorem is referenced by:  letrii  11339  nn0ge2m1nn  12541  0nelfz1  13520  fzpreddisj  13550  hashnn0n0nn  14351  hashge2el2dif  14441  divalglem5  16340  divalglem6  16341  sadcadd  16399  htpycc  24496  pco1  24531  pcohtpylem  24535  pcopt  24538  pcopt2  24539  pcoass  24540  pcorevlem  24542  vitalilem5  25129  vieta1lem2  25824  ppiltx  26681  ppiublem1  26705  chtub  26715  axlowdimlem16  28215  axlowdim  28219  lfgrnloop  28385  lfuhgr1v0e  28511  lfgrwlkprop  28944  ballotlem2  33487  subfacp1lem1  34170  subfacp1lem5  34175  bcneg1  34706  poimirlem9  36497  poimirlem16  36504  poimirlem17  36505  poimirlem19  36507  poimirlem20  36508  poimirlem22  36510  fdc  36613  pellexlem6  41572  jm2.23  41735
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