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Theorem omnimkv 6980
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 6958 . . . 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 pm2.53 694 . . . . . . 7 ((∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
43orcoms 702 . . . . . 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 1833 . . 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 6977 . 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 680  wal 1312   = wceq 1314  wcel 1463  wral 2390  wrex 2391  c0 3329  wf 5077  cfv 5081  1oc1o 6260  2oc2o 6261  Omnicomni 6954  Markovcmarkov 6975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-fn 5084  df-f 5085  df-omni 6956  df-markov 6976
This theorem is referenced by:  exmidmp  6981
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