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Theorem issod 4211
Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4189). (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 986 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 orc 686 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧))
54a1i 9 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
6 simp3r 995 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7 breq1 3902 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑧𝑦𝑅𝑧))
86, 7syl5ibcom 154 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦𝑦𝑅𝑧))
9 olc 685 . . . . . . . . . . . 12 (𝑦𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))
108, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 simp1 966 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝜑)
12 simp2r 993 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑦𝐴)
13 simp2l 992 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝐴)
14 simp3l 994 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑧𝐴)
1512, 13, 143jca 1146 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝐴𝑥𝐴𝑧𝐴))
16 potr 4200 . . . . . . . . . . . . . . . 16 ((𝑅 Po 𝐴 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
171, 16sylan 281 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
1817expcomd 1402 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧)))
1918imp 123 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2011, 15, 6, 19syl21anc 1200 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2120, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
225, 10, 213jaod 1267 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧)))
233, 22mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
24233expa 1166 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
2524expr 372 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2625ralrimiva 2482 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2726anassrs 397 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2827ralrimiva 2482 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
29 ralcom 2571 . . . 4 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3028, 29sylib 121 . . 3 ((𝜑𝑥𝐴) → ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3130ralrimiva 2482 . 2 (𝜑 → ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
32 df-iso 4189 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))))
331, 31, 32sylanbrc 413 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  w3o 946  w3a 947  wcel 1465  wral 2393   class class class wbr 3899   Po wpo 4186   Or wor 4187
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188  df-iso 4189
This theorem is referenced by:  ltsopi  7096  ltsonq  7174
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