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Theorem sotrieq 5197
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 5191 . . . . . . 7 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 688 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 pm1.2 869 . . . . . 6 ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)
42, 3nsyl 137 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐵𝐵𝑅𝐵))
5 breq2 4788 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐵𝑅𝐶))
6 breq1 4787 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝑅𝐵𝐶𝑅𝐵))
75, 6orbi12d 883 . . . . . 6 (𝐵 = 𝐶 → ((𝐵𝑅𝐵𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))
87notbid 307 . . . . 5 (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
94, 8syl5ibcom 235 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
109con2d 131 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶))
11 solin 5193 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
12 3orass 1073 . . . . . 6 ((𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
1311, 12sylib 208 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)))
14 or12 885 . . . . 5 ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1513, 14sylib 208 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶𝐶𝑅𝐵)))
1615ord 844 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶𝐶𝑅𝐵)))
1710, 16impbid 202 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶))
1817con2bid 343 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3o 1069   = wceq 1630  wcel 2144   class class class wbr 4784   Or wor 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-po 5170  df-so 5171
This theorem is referenced by:  sotrieq2  5198  sossfld  5721  soisores  6719  soisoi  6720  weniso  6746  wemapsolem  8610  distrlem4pr  10049  addcanpr  10069  sqgt0sr  10128  lttri2  10321  xrlttri2  12179  xrltne  12198  sotrine  31990  soseq  32085
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