Theorem sotritrieq 4058

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 4050	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
3	1, 2	mpan 400	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵𝑅𝐵)
4		breq2 3764	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
5	4	notbid 592	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
6	3, 5	syl5ibcom 144	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7		breq1 3763	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
8	7	notbid 592	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
9	3, 8	syl5ibcom 144	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
10	6, 9	jcad 291	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11		ioran 669	. . . 4 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
12	10, 11	syl6ibr 151	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
13	12	adantr 261	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
14		sotritric.tri	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
15		3orrot 891	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶))
16		3orcomb 894	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
17		3orass 888	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
18	15, 16, 17	3bitri 195	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
19	18	biimpi 113	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
20	19	orcomd 648	. . . 4 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
21	20	ord 643	. . 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 120	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∨ w3o 884   = wceq 1243   ∈ wcel 1393   class class class wbr 3760   Or wor 4028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-un 2919  df-sn 3378  df-pr 3379  df-op 3381  df-br 3761  df-po 4029  df-iso 4030

This theorem is referenced by:  distrlem4prl  6625  distrlem4pru  6626