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Theorem xrltnled 44642
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 11285 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
41, 2, 3syl2anc 583 1 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2098   class class class wbr 5141  *cxr 11251   < clt 11252  cle 11253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-le 11258
This theorem is referenced by:  infxrbnd2  44648  infleinflem2  44650  xrralrecnnge  44669  qinioo  44817  limsuppnflem  44995  limsupre2lem  45009  meaiuninc3v  45769  ovolval4lem1  45934  preimagelt  45984  preimalegt  45985
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