Theorem solin 5590

Description: A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)

Assertion

Ref	Expression
solin	⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))

Proof of Theorem solin

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5109	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
2		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦))
3		breq2 5110	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵))
4	1, 2, 3	3orbi123d 1464	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)))
5	4	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵))))
6		breq2 5110	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
7		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶))
8		breq1 5109	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐵 ↔ 𝐶𝑅𝐵))
9	6, 7, 8	3orbi123d 1464	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
10	9	imbi2d 343	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))))
11		df-so 5564	. . . . 5 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
12		breq1 5109	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
13		equequ1 2060	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
14		breq2 5110	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
15	12, 13, 14	3orbi123d 1464	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦𝑅𝑧)))
16	15	ralbidv 3189	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑧𝑅𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦𝑅𝑧)))
17	16	rspw 3243	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
18		breq2 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
19		equequ2 2061	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
20		breq1 5109	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
21	18, 19, 20	3orbi123d 1464	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)))
22	21	rspw 3243	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → (𝑦 ∈ 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
23	17, 22	syl6 36	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))))
24	23	impd 416	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
25	11, 24	simplbiim 514	. . . 4 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
26	25	com12 33	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
27	5, 10, 26	vtocl2ga 3541	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
28	27	impcom 413	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106   Po wpo 5561   Or wor 5562

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-so 5564

This theorem is used by:  sotric  5593  sotrieq  5594  somo  5602  wecmpep  5647  sorpssi  7769  soxp  8168  fnwe2lem4  8175  infsupprpr  9525  wemaplem2  9568  fpwwe2lem11  10753  fpwwe2lem12  10754  lttri4  11421  xmullem  13421  xmulasslem  13442  orngsqr  21153  noresle  28084  nosupbnd1lem6  28100  noinfbnd1lem6  28115  ltslin  28136  weiunso  37432  fin2so  38722  prproropf1olem4  48793