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Theorem tridc 6837
 Description: A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
Hypotheses
Ref Expression
tridc.po (𝜑𝑅 Po 𝐴)
tridc.tri (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
tridc.b (𝜑𝐵𝐴)
tridc.c (𝜑𝐶𝐴)
Assertion
Ref Expression
tridc (𝜑DECID 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
StepHypRef Expression
1 simpr 109 . . . 4 ((𝜑𝐵𝑅𝐶) → 𝐵𝑅𝐶)
21orcd 723 . . 3 ((𝜑𝐵𝑅𝐶) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
3 df-dc 821 . . 3 (DECID 𝐵𝑅𝐶 ↔ (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
42, 3sylibr 133 . 2 ((𝜑𝐵𝑅𝐶) → DECID 𝐵𝑅𝐶)
5 tridc.po . . . . . . 7 (𝜑𝑅 Po 𝐴)
6 tridc.c . . . . . . 7 (𝜑𝐶𝐴)
7 poirr 4266 . . . . . . 7 ((𝑅 Po 𝐴𝐶𝐴) → ¬ 𝐶𝑅𝐶)
85, 6, 7syl2anc 409 . . . . . 6 (𝜑 → ¬ 𝐶𝑅𝐶)
98adantr 274 . . . . 5 ((𝜑𝐵 = 𝐶) → ¬ 𝐶𝑅𝐶)
10 simpr 109 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
1110breq1d 3975 . . . . 5 ((𝜑𝐵 = 𝐶) → (𝐵𝑅𝐶𝐶𝑅𝐶))
129, 11mtbird 663 . . . 4 ((𝜑𝐵 = 𝐶) → ¬ 𝐵𝑅𝐶)
1312olcd 724 . . 3 ((𝜑𝐵 = 𝐶) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
1413, 3sylibr 133 . 2 ((𝜑𝐵 = 𝐶) → DECID 𝐵𝑅𝐶)
15 tridc.b . . . . . . 7 (𝜑𝐵𝐴)
16 po2nr 4268 . . . . . . 7 ((𝑅 Po 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → ¬ (𝐶𝑅𝐵𝐵𝑅𝐶))
175, 6, 15, 16syl12anc 1218 . . . . . 6 (𝜑 → ¬ (𝐶𝑅𝐵𝐵𝑅𝐶))
1817adantr 274 . . . . 5 ((𝜑𝐶𝑅𝐵) → ¬ (𝐶𝑅𝐵𝐵𝑅𝐶))
19 simplr 520 . . . . . 6 (((𝜑𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
20 simpr 109 . . . . . 6 (((𝜑𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
2119, 20jca 304 . . . . 5 (((𝜑𝐶𝑅𝐵) ∧ 𝐵𝑅𝐶) → (𝐶𝑅𝐵𝐵𝑅𝐶))
2218, 21mtand 655 . . . 4 ((𝜑𝐶𝑅𝐵) → ¬ 𝐵𝑅𝐶)
2322olcd 724 . . 3 ((𝜑𝐶𝑅𝐵) → (𝐵𝑅𝐶 ∨ ¬ 𝐵𝑅𝐶))
2423, 3sylibr 133 . 2 ((𝜑𝐶𝑅𝐵) → DECID 𝐵𝑅𝐶)
25 tridc.tri . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
26 breq1 3968 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
27 eqeq1 2164 . . . . 5 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
28 breq2 3969 . . . . 5 (𝑥 = 𝐵 → (𝑦𝑅𝑥𝑦𝑅𝐵))
2926, 27, 283orbi123d 1293 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵)))
30 breq2 3969 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
31 eqeq2 2167 . . . . 5 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
32 breq1 3968 . . . . 5 (𝑦 = 𝐶 → (𝑦𝑅𝐵𝐶𝑅𝐵))
3330, 31, 323orbi123d 1293 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3429, 33rspc2va 2830 . . 3 (((𝐵𝐴𝐶𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3515, 6, 25, 34syl21anc 1219 . 2 (𝜑 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
364, 14, 24, 35mpjao3dan 1289 1 (𝜑DECID 𝐵𝑅𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   ∨ w3o 962   = wceq 1335   ∈ wcel 2128  ∀wral 2435   class class class wbr 3965   Po wpo 4253 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-po 4255 This theorem is referenced by:  fimax2gtrilemstep  6838
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