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Theorem issod 4056
 Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4034). (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 924 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 orc 633 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧))
54a1i 9 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
6 simp3r 933 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7 breq1 3767 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑧𝑦𝑅𝑧))
86, 7syl5ibcom 144 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦𝑦𝑅𝑧))
9 olc 632 . . . . . . . . . . . 12 (𝑦𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))
108, 9syl6 29 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 simp1 904 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝜑)
12 simp2r 931 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑦𝐴)
13 simp2l 930 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝐴)
14 simp3l 932 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑧𝐴)
1512, 13, 143jca 1084 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝐴𝑥𝐴𝑧𝐴))
16 potr 4045 . . . . . . . . . . . . . . . 16 ((𝑅 Po 𝐴 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
171, 16sylan 267 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
1817expcomd 1330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧)))
1918imp 115 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2011, 15, 6, 19syl21anc 1134 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2120, 9syl6 29 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
225, 10, 213jaod 1199 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧)))
233, 22mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
24233expa 1104 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
2524expr 357 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2625ralrimiva 2392 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2726anassrs 380 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2827ralrimiva 2392 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
29 ralcom 2473 . . . 4 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3028, 29sylib 127 . . 3 ((𝜑𝑥𝐴) → ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3130ralrimiva 2392 . 2 (𝜑 → ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
32 df-iso 4034 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))))
331, 31, 32sylanbrc 394 1 (𝜑𝑅 Or 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629   ∨ w3o 884   ∧ w3a 885   ∈ wcel 1393  ∀wral 2306   class class class wbr 3764   Po wpo 4031   Or wor 4032 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 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034 This theorem is referenced by:  ltsopi  6418  ltsonq  6496
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