Theorem sotritrieq 4247

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 4239	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
3	1, 2	mpan 420	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵𝑅𝐵)
4		breq2 3933	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
5	4	notbid 656	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
6	3, 5	syl5ibcom 154	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7		breq1 3932	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
8	7	notbid 656	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
9	3, 8	syl5ibcom 154	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
10	6, 9	jcad 305	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11		ioran 741	. . . 4 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
12	10, 11	syl6ibr 161	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
13	12	adantr 274	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
14		sotritric.tri	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
15		3orrot 968	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶))
16		3orcomb 971	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
17		3orass 965	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
18	15, 16, 17	3bitri 205	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
19	18	biimpi 119	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
20	19	orcomd 718	. . . 4 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
21	20	ord 713	. . 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 128	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∨ w3o 961   = wceq 1331   ∈ wcel 1480   class class class wbr 3929   Or wor 4217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218  df-iso 4219

This theorem is referenced by:  distrlem4prl  7392  distrlem4pru  7393