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Theorem xrnltled 11327
Description: "Not less than" implies "less than or equal to". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
xrnltled.1 (𝜑𝐴 ∈ ℝ*)
xrnltled.2 (𝜑𝐵 ∈ ℝ*)
xrnltled.3 (𝜑 → ¬ 𝐵 < 𝐴)
Assertion
Ref Expression
xrnltled (𝜑𝐴𝐵)

Proof of Theorem xrnltled
StepHypRef Expression
1 xrnltled.3 . 2 (𝜑 → ¬ 𝐵 < 𝐴)
2 xrnltled.1 . . 3 (𝜑𝐴 ∈ ℝ*)
3 xrnltled.2 . . 3 (𝜑𝐵 ∈ ℝ*)
42, 3xrlenltd 11325 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
51, 4mpbird 257 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106   class class class wbr 5148  *cxr 11292   < clt 11293  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-le 11299
This theorem is referenced by:  supxrub  13363  infxrlb  13373  ixxub  13405  ixxlb  13406  supicclub2  13541  radcnvle  26478  xrge0infssd  32772  infxrge0lb  32775  pimxrneun  45439  liminflbuz2  45771  icccncfext  45843
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