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Theorem issod 4313
Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4291). (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 1017 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 orc 712 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧))
54a1i 9 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
6 simp3r 1026 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7 breq1 4001 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑧𝑦𝑅𝑧))
86, 7syl5ibcom 155 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦𝑦𝑅𝑧))
9 olc 711 . . . . . . . . . . . 12 (𝑦𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))
108, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 simp1 997 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝜑)
12 simp2r 1024 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑦𝐴)
13 simp2l 1023 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝐴)
14 simp3l 1025 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑧𝐴)
1512, 13, 143jca 1177 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝐴𝑥𝐴𝑧𝐴))
16 potr 4302 . . . . . . . . . . . . . . . 16 ((𝑅 Po 𝐴 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
171, 16sylan 283 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
1817expcomd 1439 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧)))
1918imp 124 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2011, 15, 6, 19syl21anc 1237 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2120, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
225, 10, 213jaod 1304 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧)))
233, 22mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
24233expa 1203 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
2524expr 375 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2625ralrimiva 2548 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2726anassrs 400 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2827ralrimiva 2548 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
29 ralcom 2638 . . . 4 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3028, 29sylib 122 . . 3 ((𝜑𝑥𝐴) → ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3130ralrimiva 2548 . 2 (𝜑 → ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
32 df-iso 4291 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))))
331, 31, 32sylanbrc 417 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  w3o 977  w3a 978  wcel 2146  wral 2453   class class class wbr 3998   Po wpo 4288   Or wor 4289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-po 4290  df-iso 4291
This theorem is referenced by:  ltsopi  7294  ltsonq  7372
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