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Theorem ltnlei 11283
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 11282 . 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 5110  cr 11057   < clt 11196  cle 11197
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-xr 11200  df-le 11202
This theorem is referenced by:  letrii  11287  nn0ge2m1nn  12489  0nelfz1  13467  fzpreddisj  13497  hashnn0n0nn  14298  hashge2el2dif  14386  divalglem5  16286  divalglem6  16287  sadcadd  16345  htpycc  24359  pco1  24394  pcohtpylem  24398  pcopt  24401  pcopt2  24402  pcoass  24403  pcorevlem  24405  vitalilem5  24992  vieta1lem2  25687  ppiltx  26542  ppiublem1  26566  chtub  26576  axlowdimlem16  27948  axlowdim  27952  lfgrnloop  28118  lfuhgr1v0e  28244  lfgrwlkprop  28677  ballotlem2  33128  subfacp1lem1  33813  subfacp1lem5  33818  bcneg1  34348  poimirlem9  36116  poimirlem16  36123  poimirlem17  36124  poimirlem19  36126  poimirlem20  36127  poimirlem22  36129  fdc  36233  pellexlem6  41186  jm2.23  41349
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