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Theorem issod 4209
Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4187). (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 984 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
4 orc 684 . . . . . . . . . . . 12 (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧))
54a1i 9 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
6 simp3r 993 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7 breq1 3900 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑧𝑦𝑅𝑧))
86, 7syl5ibcom 154 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦𝑦𝑅𝑧))
9 olc 683 . . . . . . . . . . . 12 (𝑦𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))
108, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
11 simp1 964 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝜑)
12 simp2r 991 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑦𝐴)
13 simp2l 990 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑥𝐴)
14 simp3l 992 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → 𝑧𝐴)
1512, 13, 143jca 1144 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝐴𝑥𝐴𝑧𝐴))
16 potr 4198 . . . . . . . . . . . . . . . 16 ((𝑅 Po 𝐴 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
171, 16sylan 279 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → ((𝑦𝑅𝑥𝑥𝑅𝑧) → 𝑦𝑅𝑧))
1817expcomd 1400 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧)))
1918imp 123 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐴𝑥𝐴𝑧𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2011, 15, 6, 19syl21anc 1198 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥𝑦𝑅𝑧))
2120, 9syl6 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
225, 10, 213jaod 1265 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝑅𝑦𝑦𝑅𝑧)))
233, 22mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
24233expa 1164 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝑧𝐴𝑥𝑅𝑧)) → (𝑥𝑅𝑦𝑦𝑅𝑧))
2524expr 370 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑧𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2625ralrimiva 2480 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2726anassrs 395 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ∀𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
2827ralrimiva 2480 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
29 ralcom 2569 . . . 4 (∀𝑦𝐴𝑧𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3028, 29sylib 121 . . 3 ((𝜑𝑥𝐴) → ∀𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
3130ralrimiva 2480 . 2 (𝜑 → ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧)))
32 df-iso 4187 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑧𝐴𝑦𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦𝑦𝑅𝑧))))
331, 31, 32sylanbrc 411 1 (𝜑𝑅 Or 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680  w3o 944  w3a 945  wcel 1463  wral 2391   class class class wbr 3897   Po wpo 4184   Or wor 4185
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-po 4186  df-iso 4187
This theorem is referenced by:  ltsopi  7092  ltsonq  7170
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