Theorem omnimkv 7030

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 7008	. . . 4 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))
2	1	ibi 175	. . 3 ⊢ (𝐴 ∈ Omni → ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
3		pm2.53 711	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
4	3	orcoms 719	. . . . . 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 1851	. . 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 7027	. 2 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
10	8, 9	mpbird 166	1 ⊢ (𝐴 ∈ Omni → 𝐴 ∈ Markov)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 697  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ∅c0 3363  ⟶wf 5119  ‘cfv 5123  1_oc1o 6306  2_oc2o 6307  Omnicomni 7004  Markovcmarkov 7025

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-fn 5126  df-f 5127  df-omni 7006  df-markov 7026

This theorem is referenced by:  exmidmp  7031