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Theorem ltnlei 10118
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 10117 . 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 4623  cr 9895   < clt 10034  cle 10035
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 4751  ax-nul 4759  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-xr 10038  df-le 10040
This theorem is referenced by:  letrii  10122  nn0ge2m1nn  11320  zgt1rpn0n1  11831  0nelfz1  12318  fzpreddisj  12348  hashnn0n0nn  13136  hashge2el2dif  13216  n2dvds1  15047  divalglem5  15063  divalglem6  15064  sadcadd  15123  strlemor1OLD  15909  htpycc  22719  pco1  22755  pcohtpylem  22759  pcopt  22762  pcopt2  22763  pcoass  22764  pcorevlem  22766  vitalilem5  23321  vieta1lem2  24004  ppiltx  24837  ppiublem1  24861  chtub  24871  axlowdimlem16  25771  axlowdim  25775  lfgrnloop  25949  lfuhgr1v0e  26073  lfgrwlkprop  26487  ballotlem2  30373  subfacp1lem1  30922  subfacp1lem5  30927  bcneg1  31383  poimirlem9  33089  poimirlem16  33096  poimirlem17  33097  poimirlem19  33099  poimirlem20  33100  poimirlem22  33102  fdc  33212  pellexlem6  36917  jm2.23  37082
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