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Theorem issod 4109
Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4087). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issod.1 (𝜑𝑅 Po 𝐴)
issod.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
Assertion
Ref Expression
issod (𝜑𝑅 Or 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦

Proof of Theorem issod
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 issod.1 . 2 (𝜑𝑅 Po 𝐴)
2 issod.2 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
323adant3 959 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 orc 666 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧))
54a1i 9 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
6 simp3r 968 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7 breq1 3814 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑧𝑦𝑅𝑧))
86, 7syl5ibcom 153 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦𝑦𝑅𝑧))
9 olc 665 . . . . . . . . . . . 12 (𝑦𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))
108, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 simp1 939 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝜑)
12 simp2r 966 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑦𝐴)
13 simp2l 965 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝐴)
14 simp3l 967 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑧𝐴)
1512, 13, 143jca 1119 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝐴𝑥𝐴𝑧𝐴))
16 potr 4098 . . . . . . . . . . . . . . . 16 ((𝑅 Po 𝐴 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
171, 16sylan 277 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
1817expcomd 1371 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧)))
1918imp 122 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2011, 15, 6, 19syl21anc 1169 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2120, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
225, 10, 213jaod 1236 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧)))
233, 22mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
24233expa 1139 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
2524expr 367 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2625ralrimiva 2440 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2726anassrs 392 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2827ralrimiva 2440 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
29 ralcom 2523 . . . 4 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3028, 29sylib 120 . . 3 ((𝜑𝑥𝐴) → ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3130ralrimiva 2440 . 2 (𝜑 → ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
32 df-iso 4087 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))))
331, 31, 32sylanbrc 408 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  w3o 919  w3a 920  wcel 1434  wral 2353   class class class wbr 3811   Po wpo 4084   Or wor 4085
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 2065
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-po 4086  df-iso 4087
This theorem is referenced by:  ltsopi  6781  ltsonq  6859
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