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Theorem xrlelttric 32434
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 pm2.1 893 . 2 𝐵 < 𝐴𝐵 < 𝐴)
2 xrlenlt 11276 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
32orbi1d 913 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴𝐵 < 𝐴)))
41, 3mpbiri 258 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  wcel 2098   class class class wbr 5138  *cxr 11244   < clt 11245  cle 11246
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
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 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-le 11251
This theorem is referenced by:  difioo  32462  esumpcvgval  33565
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