Theorem omniwomnimkv 7089

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.

Step	Hyp	Ref	Expression
1		elex 2720	. 2 ⊢ (𝐴 ∈ Omni → 𝐴 ∈ V)
2		simpl 108	. . 3 ⊢ ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ WOmni)
3	2	elexd 2722	. 2 ⊢ ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ V)
4		1n0 6369	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
5	4	nesymi 2370	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = 1o
6		eqeq1 2161	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) = 1o ↔ ∅ = 1o))
7	5, 6	mtbiri 665	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) = ∅ → ¬ (𝑓‘𝑥) = 1o)
8	7	reximi 2551	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ → ∃𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o)
9		rexnalim 2443	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o → ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
10	8, 9	syl 14	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ → ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
11	10	orim1i 750	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
12	11	orcomd 719	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
13		df-dc 821	. . . . . . . . 9 ⊢ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
14	12, 13	sylibr 133	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
15	14	adantl 275	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
16		simpr 109	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
17	16	orcomd 719	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
18	17	ord 714	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
19	15, 18	jca 304	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
20		simprl 521	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
21	20, 13	sylib 121	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
22		simprr 522	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
23	22	orim2d 778	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
24	21, 23	mpd 13	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
25	24	orcomd 719	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
26	19, 25	impbida 586	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) → ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
27	26	pm5.74da 440	. . . 4 ⊢ (𝐴 ∈ V → ((𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ↔ (𝑓:𝐴⟶2o → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))))
28	27	albidv 1801	. . 3 ⊢ (𝐴 ∈ V → (∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))))
29		isomni 7058	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))
30		iswomni 7087	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
31		ismkv 7075	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
32	30, 31	anbi12d 465	. . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ (∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))))
33		19.26 1458	. . . . 5 ⊢ (∀𝑓((𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) ↔ (∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
34	32, 33	bitr4di 197	. . . 4 ⊢ (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓((𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))))
35		jcab 593	. . . . 5 ⊢ ((𝑓:𝐴⟶2o → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) ↔ ((𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
36	35	albii 1447	. . . 4 ⊢ (∀𝑓(𝑓:𝐴⟶2o → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) ↔ ∀𝑓((𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
37	34, 36	bitr4di 197	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∧ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))))
38	28, 29, 37	3bitr4d 219	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov)))
39	1, 3, 38	pm5.21nii 694	1 ⊢ (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  Vcvv 2709  ∅c0 3390  ⟶wf 5159  ‘cfv 5163  1_oc1o 6346  2_oc2o 6347  Omnicomni 7056  Markovcmarkov 7073  WOmnicwomni 7085

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  ax-nul 4086

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-nul 3391  df-sn 3562  df-suc 4326  df-fn 5166  df-f 5167  df-1o 6353  df-omni 7057  df-markov 7074  df-womni 7086

This theorem is referenced by:  lpowlpo  7090