Theorem sotrieq 5598

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 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Proof of Theorem sotrieq

Step	Hyp	Ref	Expression
1		sonr 5591	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
2	1	adantrr 730	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵)
3		pm1.2 917	. . . . . 6 ⊢ ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) → 𝐵𝑅𝐵)
4	2, 3	nsyl 141	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵))
5		breq2 5111	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
6		breq1 5110	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
7	5, 6	orbi12d 932	. . . . . 6 ⊢ (𝐵 = 𝐶 → ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
8	7	notbid 321	. . . . 5 ⊢ (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
9	4, 8	syl5ibcom 248	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
10	9	con2d 135	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶))
11		solin 5594	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
12		3orass 1106	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
13	11, 12	sylib 221	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
14		or12 934	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
15	13, 14	sylib 221	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
16	15	ord 878	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
17	10, 16	impbid 215	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶))
18	17	con2bid 357	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   class class class wbr 5107   Or wor 5566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-po 5567  df-so 5568

This theorem is used by:  sotrieq2  5599  sotrine  5607  sossfld  6183  soisores  7331  soisoi  7332  weniso  7360  soseq  8160  wemapsolem  9525  distrlem4pr  11038  addcanpr  11058  sqgt0sr  11118  lttri2  11319  xrlttri2  13195  xrltne  13216  oneptri  44085