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Theorem ltnlei 10037
 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 10036 . 2 (𝐵𝐴 ↔ ¬ 𝐴 < 𝐵)
43con2bii 346 1 (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∈ wcel 1977   class class class wbr 4583  ℝcr 9814   < clt 9953   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-xr 9957  df-le 9959 This theorem is referenced by:  letrii  10041  nn0ge2m1nn  11237  zgt1rpn0n1  11747  0nelfz1  12231  fzpreddisj  12260  hashnn0n0nn  13041  hashge2el2dif  13117  n2dvds1  14942  divalglem5  14958  divalglem6  14959  sadcadd  15018  strlemor1  15796  htpycc  22587  pco1  22623  pcohtpylem  22627  pcopt  22630  pcopt2  22631  pcoass  22632  pcorevlem  22634  vitalilem5  23187  vieta1lem2  23870  ppiltx  24703  ppiublem1  24727  chtub  24737  axlowdimlem16  25637  axlowdim  25641  lfgrnloop  25791  spthispth  26103  ballotlem2  29877  subfacp1lem1  30415  subfacp1lem5  30420  bcneg1  30875  poimirlem9  32588  poimirlem16  32595  poimirlem17  32596  poimirlem19  32598  poimirlem20  32599  poimirlem22  32601  fdc  32711  pellexlem6  36416  jm2.23  36581  lfuhgr1v0e  40480  lfgrwlkprop  40896
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