Theorem sotritrieq 4465

Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)

Hypotheses

Ref	Expression
sotritric.or	⊢ 𝑅 Or 𝐴
sotritric.tri	⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))

Assertion

Ref	Expression
sotritrieq	⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Proof of Theorem sotritrieq

Step	Hyp	Ref	Expression
1		sotritric.or	. . . . . . 7 ⊢ 𝑅 Or 𝐴
2		sonr 4457	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
3	1, 2	mpan 428	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵𝑅𝐵)
4		breq2 4129	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
5	4	notbid 677	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
6	3, 5	syl5ibcom 155	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7		breq1 4128	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
8	7	notbid 677	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
9	3, 8	syl5ibcom 155	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
10	6, 9	jcad 307	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11		ioran 764	. . . 4 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
12	10, 11	imbitrrdi 162	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
13	12	adantr 276	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
14		sotritric.tri	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
15		3orrot 1015	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶))
16		3orcomb 1018	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
17		3orass 1012	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
18	15, 16, 17	3bitri 206	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
19	18	biimpi 120	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
20	19	orcomd 741	. . . 4 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
21	20	ord 736	. . 3 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) → 𝐵 = 𝐶))
22	14, 21	syl 14	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) → 𝐵 = 𝐶))
23	13, 22	impbid 129	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209   class class class wbr 4125   Or wor 4435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-po 4436  df-iso 4437

This theorem is referenced by:  distrlem4prl  7941  distrlem4pru  7942