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Theorem xrlelttric 31711
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 pm2.1 896 . 2 𝐵 < 𝐴𝐵 < 𝐴)
2 xrlenlt 11228 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
32orbi1d 916 . 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 397  wo 846  wcel 2107   class class class wbr 5109  *cxr 11196   < clt 11197  cle 11198
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-le 11203
This theorem is referenced by:  difioo  31739  esumpcvgval  32741
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