Theorem sotrieq 5570

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 5563	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
2	1	adantrr 717	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵)
3		pm1.2 903	. . . . . 6 ⊢ ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) → 𝐵𝑅𝐵)
4	2, 3	nsyl 140	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵))
5		breq2 5106	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
6		breq1 5105	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
7	5, 6	orbi12d 918	. . . . . 6 ⊢ (𝐵 = 𝐶 → ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
8	7	notbid 318	. . . . 5 ⊢ (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
9	4, 8	syl5ibcom 245	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
10	9	con2d 134	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶))
11		solin 5566	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
12		3orass 1089	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
13	11, 12	sylib 218	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
14		or12 920	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
15	13, 14	sylib 218	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
16	15	ord 864	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
17	10, 16	impbid 212	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶))
18	17	con2bid 354	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   class class class wbr 5102   Or wor 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-po 5539  df-so 5540

This theorem is referenced by:  sotrieq2  5571  sotrine  5579  sossfld  6147  soisores  7284  soisoi  7285  weniso  7311  soseq  8115  wemapsolem  9479  distrlem4pr  10955  addcanpr  10975  sqgt0sr  11035  lttri2  11232  xrlttri2  13078  xrltne  13099  oneptri  43239