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Theorem omniwomnimkv 7048
 Description: A set is omniscient if and only if it is weakly omniscient and Markov. The case says that LPO WLPO MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
Assertion
Ref Expression
omniwomnimkv Omni WOmni Markov

Proof of Theorem omniwomnimkv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2700 . 2 Omni
2 simpl 108 . . 3 WOmni Markov WOmni
32elexd 2702 . 2 WOmni Markov
4 1n0 6336 . . . . . . . . . . . . . . 15
54nesymi 2355 . . . . . . . . . . . . . 14
6 eqeq1 2147 . . . . . . . . . . . . . 14
75, 6mtbiri 665 . . . . . . . . . . . . 13
87reximi 2532 . . . . . . . . . . . 12
9 rexnalim 2428 . . . . . . . . . . . 12
108, 9syl 14 . . . . . . . . . . 11
1110orim1i 750 . . . . . . . . . 10
1211orcomd 719 . . . . . . . . 9
13 df-dc 821 . . . . . . . . 9 DECID
1412, 13sylibr 133 . . . . . . . 8 DECID
1514adantl 275 . . . . . . 7 DECID
16 simpr 109 . . . . . . . . 9
1716orcomd 719 . . . . . . . 8
1817ord 714 . . . . . . 7
1915, 18jca 304 . . . . . 6 DECID
20 simprl 521 . . . . . . . . 9 DECID DECID
2120, 13sylib 121 . . . . . . . 8 DECID
22 simprr 522 . . . . . . . . 9 DECID
2322orim2d 778 . . . . . . . 8 DECID
2421, 23mpd 13 . . . . . . 7 DECID
2524orcomd 719 . . . . . 6 DECID
2619, 25impbida 586 . . . . 5 DECID
2726pm5.74da 440 . . . 4 DECID
2827albidv 1797 . . 3 DECID
29 isomni 7015 . . 3 Omni
30 iswomni 7046 . . . . . 6 WOmni DECID
31 ismkv 7034 . . . . . 6 Markov
3230, 31anbi12d 465 . . . . 5 WOmni Markov DECID
33 19.26 1458 . . . . 5 DECID DECID
3432, 33syl6bbr 197 . . . 4 WOmni Markov DECID
35 jcab 593 . . . . 5 DECID DECID
3635albii 1447 . . . 4 DECID DECID
3734, 36syl6bbr 197 . . 3 WOmni Markov DECID
3828, 29, 373bitr4d 219 . 2 Omni WOmni Markov
391, 3, 38pm5.21nii 694 1 Omni WOmni Markov
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 698  DECID wdc 820  wal 1330   wceq 1332   wcel 1481  wral 2417  wrex 2418  cvv 2689  c0 3367  wf 5126  cfv 5130  c1o 6313  c2o 6314  Omnicomni 7011  Markovcmarkov 7032  WOmnicwomni 7044 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4061 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-nul 3368  df-sn 3537  df-suc 4300  df-fn 5133  df-f 5134  df-1o 6320  df-omni 7013  df-markov 7033  df-womni 7045 This theorem is referenced by: (None)
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