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Theorem ltnlei 10102
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 10101 . 2 (𝐵𝐴 ↔ ¬ 𝐴 < 𝐵)
43con2bii 347 1 (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 1987   class class class wbr 4613  cr 9879   < clt 10018  cle 10019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-xr 10022  df-le 10024
This theorem is referenced by:  letrii  10106  nn0ge2m1nn  11304  zgt1rpn0n1  11815  0nelfz1  12302  fzpreddisj  12332  hashnn0n0nn  13120  hashge2el2dif  13200  n2dvds1  15028  divalglem5  15044  divalglem6  15045  sadcadd  15104  strlemor1OLD  15890  htpycc  22687  pco1  22723  pcohtpylem  22727  pcopt  22730  pcopt2  22731  pcoass  22732  pcorevlem  22734  vitalilem5  23287  vieta1lem2  23970  ppiltx  24803  ppiublem1  24827  chtub  24837  axlowdimlem16  25737  axlowdim  25741  lfgrnloop  25915  lfuhgr1v0e  26039  lfgrwlkprop  26453  ballotlem2  30328  subfacp1lem1  30866  subfacp1lem5  30871  bcneg1  31327  poimirlem9  33047  poimirlem16  33054  poimirlem17  33055  poimirlem19  33057  poimirlem20  33058  poimirlem22  33060  fdc  33170  pellexlem6  36875  jm2.23  37040
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