Theorem sotritrieq 4082

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 4074	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
3	1, 2	mpan 415	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵𝑅𝐵)
4		breq2 3791	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
5	4	notbid 625	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐵𝑅𝐶))
6	3, 5	syl5ibcom 153	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐵𝑅𝐶))
7		breq1 3790	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵))
8	7	notbid 625	. . . . . 6 ⊢ (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐵 ↔ ¬ 𝐶𝑅𝐵))
9	3, 8	syl5ibcom 153	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ 𝐶𝑅𝐵))
10	6, 9	jcad 301	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
11		ioran 702	. . . 4 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
12	10, 11	syl6ibr 160	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
13	12	adantr 270	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
14		sotritric.tri	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
15		3orrot 926	. . . . . . 7 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶))
16		3orcomb 929	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐶𝑅𝐵 ∨ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
17		3orass 923	. . . . . . 7 ⊢ ((𝐵 = 𝐶 ∨ 𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
18	15, 16, 17	3bitri 204	. . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
19	18	biimpi 118	. . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
20	19	orcomd 681	. . . 4 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ∨ 𝐵 = 𝐶))
21	20	ord 676	. . 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 127	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662   ∨ w3o 919   = wceq 1285   ∈ wcel 1434   class class class wbr 3787   Or wor 4052

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-po 4053  df-iso 4054

This theorem is referenced by:  distrlem4prl  6825  distrlem4pru  6826