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Theorem xrnltled 10707
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 10705 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
51, 4mpbird 260 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2115   class class class wbr 5052  *cxr 10672   < clt 10673  cle 10674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-le 10679
This theorem is referenced by:  supxrub  12714  infxrlb  12724  ixxub  12756  ixxlb  12757  supicclub2  12891  radcnvle  25018  xrge0infssd  30496  infxrge0lb  30499  liminflbuz2  42383  icccncfext  42455
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