Theorem omnimkv 7284

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.

Step	Hyp	Ref	Expression
1		isomni 7264	. . . 4 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))
2	1	ibi 176	. . 3 ⊢ (𝐴 ∈ Omni → ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
3		pm2.53 724	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
4	3	orcoms 732	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
5	4	a1i 9	. . . . 5 ⊢ (𝐴 ∈ Omni → ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
6	5	imim2d 54	. . . 4 ⊢ (𝐴 ∈ Omni → ((𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
7	6	alimdv 1903	. . 3 ⊢ (𝐴 ∈ Omni → (∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
8	2, 7	mpd 13	. 2 ⊢ (𝐴 ∈ Omni → ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
9		ismkv 7281	. 2 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
10	8, 9	mpbird 167	1 ⊢ (𝐴 ∈ Omni → 𝐴 ∈ Markov)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 710  ∀wal 1371   = wceq 1373   ∈ wcel 2178  ∀wral 2486  ∃wrex 2487  ∅c0 3468  ⟶wf 5286  ‘cfv 5290  1_oc1o 6518  2_oc2o 6519  Omnicomni 7262  Markovcmarkov 7279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-fn 5293  df-f 5294  df-omni 7263  df-markov 7280

This theorem is referenced by:  exmidmp  7285