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Theorem xrnltled 9985
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*)
4 xrlenlt 9982 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
52, 3, 4syl2anc 691 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
61, 5mpbird 246 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wcel 1977   class class class wbr 4583  *cxr 9952   < 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-le 9959
This theorem is referenced by:  infxrlb  12036  ixxlb  12068  xrge0infssd  28916  infxrge0lb  28919  icccncfext  38773
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