Theorem omniwomnimkv 7497

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 2833	. 2 ⊢ (𝐴 ∈ Omni → 𝐴 ∈ V)
2		simpl 109	. . 3 ⊢ ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ WOmni)
3	2	elexd 2835	. 2 ⊢ ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ V)
4		1n0 6695	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
5	4	nesymi 2466	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = 1o
6		eqeq1 2245	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) = 1o ↔ ∅ = 1o))
7	5, 6	mtbiri 686	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) = ∅ → ¬ (𝑓‘𝑥) = 1o)
8	7	reximi 2647	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ → ∃𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o)
9		rexnalim 2539	. . . . . . . . . . . 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 772	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
12	11	orcomd 741	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
13		df-dc 847	. . . . . . . . 9 ⊢ (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ∨ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
14	12, 13	sylibr 134	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
15	14	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
16		simpr 110	. . . . . . . . 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 741	. . . . . . . 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 736	. . . . . . 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 306	. . . . . 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 535	. . . . . . . . 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 122	. . . . . . . 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 537	. . . . . . . . 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 800	. . . . . . . 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 741	. . . . . 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 604	. . . . 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 447	. . . 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 1877	. . 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 7466	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))
30		iswomni 7495	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
31		ismkv 7483	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
32	30, 31	anbi12d 477	. . . . 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 1534	. . . . 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 198	. . . 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 611	. . . . 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 1523	. . . 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 198	. . 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 220	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov)))
39	1, 3, 38	pm5.21nii 716	1 ⊢ (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821  ∅c0 3520  ⟶wf 5368  ‘cfv 5372  1_oc1o 6670  2_oc2o 6671  Omnicomni 7464  Markovcmarkov 7481  WOmnicwomni 7493

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3711  df-suc 4511  df-fn 5375  df-f 5376  df-1o 6677  df-omni 7465  df-markov 7482  df-womni 7494

This theorem is referenced by:  lpowlpo  7498