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Theorem omnimkv 7078
 Description: An omniscient set is Markov. In particular, the case where 𝐴 is ω means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
Assertion
Ref Expression
omnimkv (𝐴 ∈ Omni → 𝐴 ∈ Markov)

Proof of Theorem omnimkv
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isomni 7058 . . . 4 (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
21ibi 175 . . 3 (𝐴 ∈ Omni → ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)))
3 pm2.53 712 . . . . . . 7 ((∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
43orcoms 720 . . . . . 6 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
54a1i 9 . . . . 5 (𝐴 ∈ Omni → ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))
65imim2d 54 . . . 4 (𝐴 ∈ Omni → ((𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
76alimdv 1856 . . 3 (𝐴 ∈ Omni → (∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
82, 7mpd 13 . 2 (𝐴 ∈ Omni → ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))
9 ismkv 7075 . 2 (𝐴 ∈ Omni → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
108, 9mpbird 166 1 (𝐴 ∈ Omni → 𝐴 ∈ Markov)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  ∅c0 3390  ⟶wf 5159  ‘cfv 5163  1oc1o 6346  2oc2o 6347  Omnicomni 7056  Markovcmarkov 7073 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-fn 5166  df-f 5167  df-omni 7057  df-markov 7074 This theorem is referenced by:  exmidmp  7079
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