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Theorem xrnltled 11253
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 11250 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
51, 4mpbird 259 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2144   class class class wbr 5102  *cxr 11217   < clt 11218  cle 11219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-le 11224
This theorem is referenced by:  supxrub  13329  infxrlb  13340  ixxub  13372  ixxlb  13373  supicclub2  13510  radcnvle  26485  xrge0infssd  32965  infxrge0lb  32968  pimxrneun  46067  liminflbuz2  46394  icccncfext  46466
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