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Theorem ltnlei 10754
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 10753 . 2 (𝐵𝐴 ↔ ¬ 𝐴 < 𝐵)
43con2bii 361 1 (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wcel 2112   class class class wbr 5033  cr 10529   < clt 10668  cle 10669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-xr 10672  df-le 10674
This theorem is referenced by:  letrii  10758  nn0ge2m1nn  11956  0nelfz1  12925  fzpreddisj  12955  hashnn0n0nn  13752  hashge2el2dif  13838  divalglem5  15741  divalglem6  15742  sadcadd  15800  htpycc  23588  pco1  23623  pcohtpylem  23627  pcopt  23630  pcopt2  23631  pcoass  23632  pcorevlem  23634  vitalilem5  24219  vieta1lem2  24910  ppiltx  25765  ppiublem1  25789  chtub  25799  axlowdimlem16  26754  axlowdim  26758  lfgrnloop  26921  lfuhgr1v0e  27047  lfgrwlkprop  27480  ballotlem2  31854  subfacp1lem1  32534  subfacp1lem5  32539  bcneg1  33076  poimirlem9  35059  poimirlem16  35066  poimirlem17  35067  poimirlem19  35069  poimirlem20  35070  poimirlem22  35072  fdc  35176  pellexlem6  39762  jm2.23  39924
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