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Theorem sotrieq 5495
Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
sotrieq ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Proof of Theorem sotrieq
StepHypRef Expression
1 sonr 5489 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 713 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 pm1.2 897 . . . . . 6 ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)
42, 3nsyl 142 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐵𝐵𝑅𝐵))
5 breq2 5061 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
6 breq1 5060 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
75, 6orbi12d 912 . . . . . 6 (𝐵 = 𝐶 → ((𝐵𝑅𝐵𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))
87notbid 319 . . . . 5 (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
94, 8syl5ibcom 246 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
109con2d 136 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶))
11 solin 5491 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
12 3orass 1082 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1311, 12sylib 219 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
14 or12 914 . . . . 5 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1513, 14sylib 219 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1615ord 858 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶𝐶𝑅𝐵)))
1710, 16impbid 213 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶))
1817con2bid 356 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3o 1078   = wceq 1528  wcel 2105   class class class wbr 5057   Or wor 5466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-po 5467  df-so 5468
This theorem is referenced by:  sotrieq2  5496  sossfld  6036  soisores  7069  soisoi  7070  weniso  7096  wemapsolem  9002  distrlem4pr  10436  addcanpr  10456  sqgt0sr  10516  lttri2  10711  xrlttri2  12523  xrltne  12544  sotrine  32900  soseq  32993
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