Theorem tridc 6786

Description: A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)

Hypotheses

Ref	Expression
tridc.po	⊢ (𝜑 → 𝑅 Po 𝐴)
tridc.tri	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
tridc.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
tridc.c	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
tridc	⊢ (𝜑 → DECID 𝐵𝑅𝐶)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝑅,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem tridc

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
2	1	orcd 722	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
3		df-dc 820	. . 3 ⊢ (DECID 𝐵𝑅𝐶 ↔ (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
4	2, 3	sylibr 133	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → DECID 𝐵𝑅𝐶)
5		tridc.po	. . . . . . 7 ⊢ (𝜑 → 𝑅 Po 𝐴)
6		tridc.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
7		poirr 4224	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
8	5, 6, 7	syl2anc 408	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶𝑅𝐶)
9	8	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ¬ 𝐶𝑅𝐶)
10		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
11	10	breq1d 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵𝑅𝐶 ↔ 𝐶𝑅𝐶))
12	9, 11	mtbird 662	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → ¬ 𝐵𝑅𝐶)
13	12	olcd 723	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
14	13, 3	sylibr 133	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → DECID 𝐵𝑅𝐶)
15		tridc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
16		po2nr 4226	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ¬ (𝐶𝑅𝐵 ∧ 𝐵𝑅𝐶))
17	5, 6, 15, 16	syl12anc 1214	. . . . . 6 ⊢ (𝜑 → ¬ (𝐶𝑅𝐵 ∧ 𝐵𝑅𝐶))
18	17	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → ¬ (𝐶𝑅𝐵 ∧ 𝐵𝑅𝐶))
19		simplr 519	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
20		simpr 109	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
21	19, 20	jca 304	. . . . 5 ⊢ (((𝜑 ∧ 𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → (𝐶𝑅𝐵 ∧ 𝐵𝑅𝐶))
22	18, 21	mtand 654	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → ¬ 𝐵𝑅𝐶)
23	22	olcd 723	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
24	23, 3	sylibr 133	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → DECID 𝐵𝑅𝐶)
25		tridc.tri	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
26		breq1 3927	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
27		eqeq1 2144	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦))
28		breq2 3928	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵))
29	26, 27, 28	3orbi123d 1289	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵)))
30		breq2 3928	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
31		eqeq2 2147	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶))
32		breq1 3927	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐵 ↔ 𝐶𝑅𝐵))
33	30, 31, 32	3orbi123d 1289	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∨ 𝐵 = 𝑦 ∨ 𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
34	29, 33	rspc2va 2798	. . 3 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
35	15, 6, 25, 34	syl21anc 1215	. 2 ⊢ (𝜑 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
36	4, 14, 24, 35	mpjao3dan 1285	1 ⊢ (𝜑 → DECID 𝐵𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   ∨ w3o 961   = wceq 1331   ∈ wcel 1480  ∀wral 2414   class class class wbr 3924   Po wpo 4211

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 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-po 4213

This theorem is referenced by:  fimax2gtrilemstep  6787