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Theorem xrltnled 40323
Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
xrltnled.1 (𝜑𝐴 ∈ ℝ*)
xrltnled.2 (𝜑𝐵 ∈ ℝ*)
Assertion
Ref Expression
xrltnled (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Proof of Theorem xrltnled
StepHypRef Expression
1 xrltnled.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 xrltnled.2 . 2 (𝜑𝐵 ∈ ℝ*)
3 xrltnle 10395 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
41, 2, 3syl2anc 580 1 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wcel 2157   class class class wbr 4843  *cxr 10362   < clt 10363  cle 10364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-le 10369
This theorem is referenced by:  infxrbnd2  40329  infleinflem2  40331  xrralrecnnge  40356  qinioo  40506  limsuppnflem  40686  limsupre2lem  40700  meaiuninc3v  41444  ovolval4lem1  41609  preimagelt  41658  preimalegt  41659
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