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Theorem omniwomnimkv 7048
Description: A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω 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 (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))

Proof of Theorem omniwomnimkv
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2700 . 2 (𝐴 ∈ Omni → 𝐴 ∈ V)
2 simpl 108 . . 3 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ WOmni)
32elexd 2702 . 2 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ V)
4 1n0 6336 . . . . . . . . . . . . . . 15 1o ≠ ∅
54nesymi 2355 . . . . . . . . . . . . . 14 ¬ ∅ = 1o
6 eqeq1 2147 . . . . . . . . . . . . . 14 ((𝑓𝑥) = ∅ → ((𝑓𝑥) = 1o ↔ ∅ = 1o))
75, 6mtbiri 665 . . . . . . . . . . . . 13 ((𝑓𝑥) = ∅ → ¬ (𝑓𝑥) = 1o)
87reximi 2532 . . . . . . . . . . . 12 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ∃𝑥𝐴 ¬ (𝑓𝑥) = 1o)
9 rexnalim 2428 . . . . . . . . . . . 12 (∃𝑥𝐴 ¬ (𝑓𝑥) = 1o → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
108, 9syl 14 . . . . . . . . . . 11 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
1110orim1i 750 . . . . . . . . . 10 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1211orcomd 719 . . . . . . . . 9 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
13 df-dc 821 . . . . . . . . 9 (DECID𝑥𝐴 (𝑓𝑥) = 1o ↔ (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1412, 13sylibr 133 . . . . . . . 8 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
1514adantl 275 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
16 simpr 109 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1716orcomd 719 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅))
1817ord 714 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
1915, 18jca 304 . . . . . 6 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))
20 simprl 521 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
2120, 13sylib 121 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
22 simprr 522 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
2322orim2d 778 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → ((∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅)))
2421, 23mpd 13 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅))
2524orcomd 719 . . . . . 6 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
2619, 25impbida 586 . . . . 5 ((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) → ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) ↔ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
2726pm5.74da 440 . . . 4 (𝐴 ∈ V → ((𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) ↔ (𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
2827albidv 1797 . . 3 (𝐴 ∈ V → (∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
29 isomni 7015 . . 3 (𝐴 ∈ V → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
30 iswomni 7046 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o)))
31 ismkv 7034 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3230, 31anbi12d 465 . . . . 5 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ (∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
33 19.26 1458 . . . . 5 (∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ (∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3432, 33syl6bbr 197 . . . 4 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
35 jcab 593 . . . . 5 ((𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ ((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3635albii 1447 . . . 4 (∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ ∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3734, 36syl6bbr 197 . . 3 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
3828, 29, 373bitr4d 219 . 2 (𝐴 ∈ V → (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov)))
391, 3, 38pm5.21nii 694 1 (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ 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  Vcvv 2689  c0 3367  wf 5126  cfv 5130  1oc1o 6313  2oc2o 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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