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Theorem xrltnled 44774
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 11319 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
41, 2, 3syl2anc 582 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 5152  *cxr 11285   < clt 11286  cle 11287
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-le 11292
This theorem is referenced by:  infxrbnd2  44780  infleinflem2  44782  xrralrecnnge  44801  qinioo  44949  limsuppnflem  45127  limsupre2lem  45141  meaiuninc3v  45901  ovolval4lem1  46066  preimagelt  46116  preimalegt  46117
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