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Theorem xrltnled 43946
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 11268 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
41, 2, 3syl2anc 585 1 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2107   class class class wbr 5144  *cxr 11234   < clt 11235  cle 11236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-cnv 5680  df-le 11241
This theorem is referenced by:  infxrbnd2  43952  infleinflem2  43954  xrralrecnnge  43973  qinioo  44121  limsuppnflem  44299  limsupre2lem  44313  meaiuninc3v  45073  ovolval4lem1  45238  preimagelt  45288  preimalegt  45289
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