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Theorem xrltnled 39892
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 10143 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
41, 2, 3syl2anc 694 1 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 2030   class class class wbr 4685  *cxr 10111   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-le 10118
This theorem is referenced by:  infxrbnd2  39898  infleinflem2  39900  xrralrecnnge  39926  qinioo  40080  limsuppnflem  40260  limsupre2lem  40274  meaiuninc3v  41019  ovolval4lem1  41184  preimagelt  41233  preimalegt  41234
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