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Theorem xrlelttric 30063
Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
Assertion
Ref Expression
xrlelttric ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Proof of Theorem xrlelttric
StepHypRef Expression
1 pm2.1 927 . 2 𝐵 < 𝐴𝐵 < 𝐴)
2 xrlenlt 10421 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
32orbi1d 947 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴𝐵 < 𝐴)))
41, 3mpbiri 250 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 880  wcel 2166   class class class wbr 4872  *cxr 10389   < clt 10390  cle 10391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347  df-cnv 5349  df-le 10396
This theorem is referenced by:  difioo  30090  esumpcvgval  30684
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